diff --git a/specs/phase0/fork-choice.md b/specs/phase0/fork-choice.md
index 0b14297ad..8e1f65121 100644
--- a/specs/phase0/fork-choice.md
+++ b/specs/phase0/fork-choice.md
@@ -303,7 +303,8 @@ def store_target_checkpoint_state(store: Store, target: Checkpoint) -> None:
     # Store target checkpoint state if not yet seen
     if target not in store.checkpoint_states:
         base_state = copy(store.block_states[target.root])
-        process_slots(base_state, compute_start_slot_at_epoch(target.epoch))
+        if base_state.slot < compute_start_slot_at_epoch(target.epoch):
+            process_slots(base_state, compute_start_slot_at_epoch(target.epoch))
         store.checkpoint_states[target] = base_state
 ```
 
diff --git a/specs/phase1/beacon-chain.md b/specs/phase1/beacon-chain.md
index a1a940adf..e103d51f4 100644
--- a/specs/phase1/beacon-chain.md
+++ b/specs/phase1/beacon-chain.md
@@ -569,7 +569,7 @@ def get_light_client_committee(beacon_state: BeaconState, epoch: Epoch) -> Seque
         seed=seed,
         index=0,
         count=get_active_shard_count(beacon_state),
-    )[:TARGET_COMMITTEE_SIZE]
+    )[:LIGHT_CLIENT_COMMITTEE_SIZE]
 ```
 
 #### `get_shard_proposer_index`
@@ -652,11 +652,11 @@ def get_offset_slots(state: BeaconState, shard: Shard) -> Sequence[Slot]:
 
 ```python
 def is_on_time_attestation(state: BeaconState,
-                           attestation: Attestation) -> bool:
+                           attestation_data: AttestationData) -> bool:
     """
-    Check if the given attestation is on-time.
+    Check if the given ``attestation_data`` is on-time.
     """
-    return attestation.data.slot == compute_previous_slot(state.slot)
+    return attestation_data.slot == compute_previous_slot(state.slot)
 ```
 
 #### `is_winning_attestation`
@@ -667,11 +667,11 @@ def is_winning_attestation(state: BeaconState,
                            committee_index: CommitteeIndex,
                            winning_root: Root) -> bool:
     """
-    Check if ``attestation`` helped contribute to the successful crosslink of
-    ``winning_root`` formed by ``committee_index`` committee at the current slot.
+    Check if on-time ``attestation`` helped contribute to the successful crosslink of
+    ``winning_root`` formed by ``committee_index`` committee.
     """
     return (
-        attestation.data.slot == state.slot
+        is_on_time_attestation(state, attestation.data)
         and attestation.data.index == committee_index
         and attestation.data.shard_transition_root == winning_root
     )
@@ -766,12 +766,14 @@ def validate_attestation(state: BeaconState, attestation: Attestation) -> None:
         assert attestation.data.source == state.previous_justified_checkpoint
 
     # Type 1: on-time attestations
-    if is_on_time_attestation(state, attestation):
+    if is_on_time_attestation(state, attestation.data):
         # Correct parent block root
         assert data.beacon_block_root == get_block_root_at_slot(state, compute_previous_slot(state.slot))
         # Correct shard number
         shard = compute_shard_from_committee_index(state, attestation.data.index, attestation.data.slot)
         assert attestation.data.shard == shard
+        # On-time attestations should have a non-empty shard transition root
+        assert attestation.data.shard_transition_root != hash_tree_root(ShardTransition())
     # Type 2: no shard transition
     else:
         # Ensure delayed attestation
@@ -886,9 +888,6 @@ def process_crosslink_for_shard(state: BeaconState,
         for attestation in transition_attestations:
             participants = get_attesting_indices(state, attestation.data, attestation.aggregation_bits)
             transition_participants = transition_participants.union(participants)
-            assert attestation.data.shard_head_root == shard_transition.shard_data_roots[
-                len(shard_transition.shard_data_roots) - 1
-            ]
 
         enough_online_stake = (
             get_total_balance(state, online_indices.intersection(transition_participants)) * 3 >=
@@ -901,6 +900,12 @@ def process_crosslink_for_shard(state: BeaconState,
         # Attestation <-> shard transition consistency
         assert shard_transition_root == hash_tree_root(shard_transition)
 
+        # Check `shard_head_root` of the winning root
+        last_offset_index = len(shard_transition.shard_states) - 1
+        shard_head_root = shard_transition.shard_states[last_offset_index].latest_block_root
+        for attestation in transition_attestations:
+            assert attestation.data.shard_head_root == shard_head_root
+
         # Apply transition
         apply_shard_transition(state, shard, shard_transition)
         # Apply proposer reward and cost
@@ -939,7 +944,7 @@ def process_crosslinks(state: BeaconState,
         # Since the attestations are validated, all `shard_attestations` satisfy `attestation.data.shard == shard`
         shard_attestations = [
             attestation for attestation in attestations
-            if is_on_time_attestation(state, attestation) and attestation.data.index == committee_index
+            if is_on_time_attestation(state, attestation.data) and attestation.data.index == committee_index
         ]
         winning_root = process_crosslink_for_shard(
             state, committee_index, shard_transitions[shard], shard_attestations
diff --git a/specs/phase1/phase1-fork.md b/specs/phase1/phase1-fork.md
index 496191e07..e83e9ef4a 100644
--- a/specs/phase1/phase1-fork.md
+++ b/specs/phase1/phase1-fork.md
@@ -111,7 +111,7 @@ def upgrade_to_phase1(pre: phase0.BeaconState) -> BeaconState:
         current_light_committee=CompactCommittee(),  # computed after state creation
         next_light_committee=CompactCommittee(),
         # Custody game
-        exposed_derived_secrets=[] * EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS,
+        exposed_derived_secrets=[()] * EARLY_DERIVED_SECRET_PENALTY_MAX_FUTURE_EPOCHS,
         # exposed_derived_secrets will fully default to zeroes
     )
     next_epoch = Epoch(epoch + 1)
diff --git a/specs/phase1/shard-fork-choice.md b/specs/phase1/shard-fork-choice.md
index 411ad9b6b..c34501a6d 100644
--- a/specs/phase1/shard-fork-choice.md
+++ b/specs/phase1/shard-fork-choice.md
@@ -36,7 +36,7 @@ This document is the shard chain fork choice spec for part of Ethereum 2.0 Phase
 @dataclass
 class ShardStore:
     shard: Shard
-    blocks: Dict[Root, ShardBlock] = field(default_factory=dict)
+    signed_blocks: Dict[Root, SignedShardBlock] = field(default_factory=dict)
     block_states: Dict[Root, ShardState] = field(default_factory=dict)
 ```
 
@@ -46,7 +46,11 @@ class ShardStore:
 def get_forkchoice_shard_store(anchor_state: BeaconState, shard: Shard) -> ShardStore:
     return ShardStore(
         shard=shard,
-        blocks={anchor_state.shard_states[shard].latest_block_root: ShardBlock(slot=anchor_state.slot, shard=shard)},
+        signed_blocks={
+            anchor_state.shard_states[shard].latest_block_root: SignedShardBlock(
+                message=ShardBlock(slot=anchor_state.slot, shard=shard)
+            )
+        },
         block_states={anchor_state.shard_states[shard].latest_block_root: anchor_state.copy().shard_states[shard]},
     )
 ```
@@ -65,7 +69,7 @@ def get_shard_latest_attesting_balance(store: Store, shard_store: ShardStore, ro
             # would be ignored once their newer vote is accepted. Check if it makes sense.
             and store.latest_messages[i].shard == shard_store.shard
             and get_shard_ancestor(
-                store, shard_store, store.latest_messages[i].shard_root, shard_store.blocks[root].slot
+                store, shard_store, store.latest_messages[i].shard_root, shard_store.signed_blocks[root].message.slot
             ) == root
         )
     ))
@@ -80,8 +84,8 @@ def get_shard_head(store: Store, shard_store: ShardStore) -> Root:
     shard_head_state = store.block_states[beacon_head_root].shard_states[shard_store.shard]
     shard_head_root = shard_head_state.latest_block_root
     shard_blocks = {
-        root: shard_block for root, shard_block in shard_store.blocks.items()
-        if shard_block.slot > shard_head_state.slot
+        root: signed_shard_block.message for root, signed_shard_block in shard_store.signed_blocks.items()
+        if signed_shard_block.message.slot > shard_head_state.slot
     }
     while True:
         # Find the valid child block roots
@@ -101,7 +105,7 @@ def get_shard_head(store: Store, shard_store: ShardStore) -> Root:
 
 ```python
 def get_shard_ancestor(store: Store, shard_store: ShardStore, root: Root, slot: Slot) -> Root:
-    block = shard_store.blocks[root]
+    block = shard_store.signed_blocks[root].message
     if block.slot > slot:
         return get_shard_ancestor(store, shard_store, block.shard_parent_root, slot)
     elif block.slot == slot:
@@ -114,7 +118,7 @@ def get_shard_ancestor(store: Store, shard_store: ShardStore, root: Root, slot:
 #### `get_pending_shard_blocks`
 
 ```python
-def get_pending_shard_blocks(store: Store, shard_store: ShardStore) -> Sequence[ShardBlock]:
+def get_pending_shard_blocks(store: Store, shard_store: ShardStore) -> Sequence[SignedShardBlock]:
     """
     Return the canonical shard block branch that has not yet been crosslinked.
     """
@@ -126,14 +130,14 @@ def get_pending_shard_blocks(store: Store, shard_store: ShardStore) -> Sequence[
 
     shard_head_root = get_shard_head(store, shard_store)
     root = shard_head_root
-    shard_blocks = []
+    signed_shard_blocks = []
     while root != latest_shard_block_root:
-        shard_block = shard_store.blocks[root]
-        shard_blocks.append(shard_block)
-        root = shard_block.shard_parent_root
+        signed_shard_block = shard_store.signed_blocks[root]
+        signed_shard_blocks.append(signed_shard_block)
+        root = signed_shard_block.message.shard_parent_root
 
-    shard_blocks.reverse()
-    return shard_blocks
+    signed_shard_blocks.reverse()
+    return signed_shard_blocks
 ```
 
 ### Handlers
@@ -169,14 +173,15 @@ def on_shard_block(store: Store, shard_store: ShardStore, signed_shard_block: Si
     )
 
     # Check the block is valid and compute the post-state
-    assert verify_shard_block_message(beacon_parent_state, shard_parent_state, shard_block)
-    assert verify_shard_block_signature(beacon_parent_state, signed_shard_block)
-
-    post_state = get_post_shard_state(shard_parent_state, shard_block)
+    shard_state = shard_parent_state.copy()
+    shard_state_transition(
+        shard_state, signed_shard_block,
+        validate=True, beacon_parent_state=beacon_parent_state)
 
     # Add new block to the store
-    shard_store.blocks[hash_tree_root(shard_block)] = shard_block
+    # Note: storing `SignedShardBlock` format for computing `ShardTransition.proposer_signature_aggregate` 
+    shard_store.signed_blocks[hash_tree_root(shard_block)] = signed_shard_block
 
     # Add new state for this block to the store
-    shard_store.block_states[hash_tree_root(shard_block)] = post_state
+    shard_store.block_states[hash_tree_root(shard_block)] = shard_state
 ```
diff --git a/specs/phase1/shard-transition.md b/specs/phase1/shard-transition.md
index 50694ce68..05f986000 100644
--- a/specs/phase1/shard-transition.md
+++ b/specs/phase1/shard-transition.md
@@ -42,8 +42,6 @@ def verify_shard_block_message(beacon_parent_state: BeaconState,
     next_slot = Slot(block.slot + 1)
     offset_slots = compute_offset_slots(get_latest_slot_for_shard(beacon_parent_state, shard), next_slot)
     assert block.slot in offset_slots
-    # Check `shard` field
-    assert block.shard == shard
     # Check `proposer_index` field
     assert block.proposer_index == get_shard_proposer_index(beacon_parent_state, block.slot, shard)
     # Check `body` field
@@ -52,10 +50,10 @@ def verify_shard_block_message(beacon_parent_state: BeaconState,
 ```
 
 ```python
-def verify_shard_block_signature(beacon_state: BeaconState,
+def verify_shard_block_signature(beacon_parent_state: BeaconState,
                                  signed_block: SignedShardBlock) -> bool:
-    proposer = beacon_state.validators[signed_block.message.proposer_index]
-    domain = get_domain(beacon_state, DOMAIN_SHARD_PROPOSAL, compute_epoch_at_slot(signed_block.message.slot))
+    proposer = beacon_parent_state.validators[signed_block.message.proposer_index]
+    domain = get_domain(beacon_parent_state, DOMAIN_SHARD_PROPOSAL, compute_epoch_at_slot(signed_block.message.slot))
     signing_root = compute_signing_root(signed_block.message, domain)
     return bls.Verify(proposer.pubkey, signing_root, signed_block.signature)
 ```
@@ -64,7 +62,21 @@ def verify_shard_block_signature(beacon_state: BeaconState,
 
 ```python
 def shard_state_transition(shard_state: ShardState,
-                           block: ShardBlock) -> None:
+                           signed_block: SignedShardBlock,
+                           validate: bool = True,
+                           beacon_parent_state: Optional[BeaconState] = None) -> ShardState:
+    if validate:
+        assert beacon_parent_state is not None
+        assert verify_shard_block_message(beacon_parent_state, shard_state, signed_block.message)
+        assert verify_shard_block_signature(beacon_parent_state, signed_block)
+
+    process_shard_block(shard_state, signed_block.message)
+    return shard_state
+```
+
+```python
+def process_shard_block(shard_state: ShardState,
+                        block: ShardBlock) -> None:
     """
     Update ``shard_state`` with shard ``block``.
     """
@@ -79,19 +91,6 @@ def shard_state_transition(shard_state: ShardState,
     shard_state.latest_block_root = latest_block_root
 ```
 
-We have a pure function `get_post_shard_state` for describing the fraud proof verification and honest validator behavior.
-
-```python
-def get_post_shard_state(shard_state: ShardState,
-                         block: ShardBlock) -> ShardState:
-    """
-    A pure function that returns a new post ShardState instead of modifying the given `shard_state`.
-    """
-    post_state = shard_state.copy()
-    shard_state_transition(post_state, block)
-    return post_state
-```
-
 ## Fraud proofs
 
 ### Verifying the proof
@@ -129,7 +128,7 @@ def is_valid_fraud_proof(beacon_state: BeaconState,
     else:
         shard_state = transition.shard_states[offset_index - 1]  # Not doing the actual state updates here.
 
-    shard_state = get_post_shard_state(shard_state, block)
+    shard_state_transition(shard_state, block, validate=False)
     if shard_state != transition.shard_states[offset_index]:
         return True
 
diff --git a/specs/phase1/validator.md b/specs/phase1/validator.md
index f347f8757..8b7ade60b 100644
--- a/specs/phase1/validator.md
+++ b/specs/phase1/validator.md
@@ -157,7 +157,7 @@ def get_shard_winning_roots(state: BeaconState,
         # All attestations in the block for this committee/shard and are "on time"
         shard_attestations = [
             attestation for attestation in attestations
-            if is_on_time_attestation(state, attestation) and attestation.data.index == committee_index
+            if is_on_time_attestation(state, attestation.data) and attestation.data.index == committee_index
         ]
         committee = get_beacon_committee(state, on_time_attestation_slot, committee_index)
 
@@ -280,7 +280,6 @@ def get_shard_transition_fields(
     beacon_state: BeaconState,
     shard: Shard,
     shard_blocks: Sequence[SignedShardBlock],
-    validate_signature: bool=True,
 ) -> Tuple[Sequence[uint64], Sequence[Root], Sequence[ShardState]]:
     shard_states = []
     shard_data_roots = []
@@ -299,7 +298,8 @@ def get_shard_transition_fields(
         else:
             shard_block = SignedShardBlock(message=ShardBlock(slot=slot, shard=shard))
             shard_data_roots.append(Root())
-        shard_state = get_post_shard_state(shard_state, shard_block.message)
+        shard_state = shard_state.copy()
+        shard_state_transition(shard_state, shard_block, validate=False)
         shard_states.append(shard_state)
         shard_block_lengths.append(len(shard_block.message.body))
 
diff --git a/ssz/simple-serialize.md b/ssz/simple-serialize.md
index 700a428e8..a9d1da56e 100644
--- a/ssz/simple-serialize.md
+++ b/ssz/simple-serialize.md
@@ -251,7 +251,7 @@ We similarly define "summary types" and "expansion types". For example, [`Beacon
 | Rust | Lighthouse | Sigma Prime | [https://github.com/sigp/lighthouse/tree/master/eth2/utils/ssz](https://github.com/sigp/lighthouse/tree/master/eth2/utils/ssz) |
 | Nim | Nimbus | Status | [https://github.com/status-im/nim-beacon-chain/blob/master/beacon_chain/ssz.nim](https://github.com/status-im/nim-beacon-chain/blob/master/beacon_chain/ssz.nim) |
 | Rust | Shasper | ParityTech | [https://github.com/paritytech/shasper/tree/master/utils/ssz](https://github.com/paritytech/shasper/tree/master/utils/ssz) |
-| TypeScript | Lodestar | ChainSafe Systems | [https://github.com/ChainSafe/ssz-js](https://github.com/ChainSafe/ssz-js) |
+| TypeScript | Lodestar | ChainSafe Systems | [https://github.com/ChainSafe/ssz-js](https://github.com/ChainSafe/ssz) |
 | Java | Cava | ConsenSys | [https://www.github.com/ConsenSys/cava/tree/master/ssz](https://www.github.com/ConsenSys/cava/tree/master/ssz) |
 | Go | Prysm | Prysmatic Labs | [https://github.com/prysmaticlabs/go-ssz](https://github.com/prysmaticlabs/go-ssz) |
 | Swift | Yeeth | Dean Eigenmann | [https://github.com/yeeth/SimpleSerialize.swift](https://github.com/yeeth/SimpleSerialize.swift) |
diff --git a/tests/core/pyspec/eth2spec/test/fork_choice/test_on_attestation.py b/tests/core/pyspec/eth2spec/test/fork_choice/test_on_attestation.py
index c1a5b2b33..ffd8b417d 100644
--- a/tests/core/pyspec/eth2spec/test/fork_choice/test_on_attestation.py
+++ b/tests/core/pyspec/eth2spec/test/fork_choice/test_on_attestation.py
@@ -158,9 +158,9 @@ def test_on_attestation_inconsistent_target_and_head(spec, state):
 
 @with_all_phases
 @spec_state_test
-def test_on_attestation_target_not_in_store(spec, state):
+def test_on_attestation_target_block_not_in_store(spec, state):
     store = spec.get_forkchoice_store(state)
-    time = store.time + spec.SECONDS_PER_SLOT * spec.SLOTS_PER_EPOCH
+    time = store.time + spec.SECONDS_PER_SLOT * (spec.SLOTS_PER_EPOCH + 1)
     spec.on_tick(store, time)
 
     # move to immediately before next epoch to make block new target
@@ -178,11 +178,63 @@ def test_on_attestation_target_not_in_store(spec, state):
     run_on_attestation(spec, state, store, attestation, False)
 
 
+@with_all_phases
+@spec_state_test
+def test_on_attestation_target_checkpoint_not_in_store(spec, state):
+    store = spec.get_forkchoice_store(state)
+    time = store.time + spec.SECONDS_PER_SLOT * (spec.SLOTS_PER_EPOCH + 1)
+    spec.on_tick(store, time)
+
+    # move to immediately before next epoch to make block new target
+    next_epoch = spec.get_current_epoch(state) + 1
+    transition_to(spec, state, spec.compute_start_slot_at_epoch(next_epoch) - 1)
+
+    target_block = build_empty_block_for_next_slot(spec, state)
+    signed_target_block = state_transition_and_sign_block(spec, state, target_block)
+
+    # add target block to store
+    spec.on_block(store, signed_target_block)
+
+    # target checkpoint state is not yet in store
+
+    attestation = get_valid_attestation(spec, state, slot=target_block.slot, signed=True)
+    assert attestation.data.target.root == target_block.hash_tree_root()
+
+    run_on_attestation(spec, state, store, attestation)
+
+
+@with_all_phases
+@spec_state_test
+def test_on_attestation_target_checkpoint_not_in_store_diff_slot(spec, state):
+    store = spec.get_forkchoice_store(state)
+    time = store.time + spec.SECONDS_PER_SLOT * (spec.SLOTS_PER_EPOCH + 1)
+    spec.on_tick(store, time)
+
+    # move to two slots before next epoch to make target block one before an empty slot
+    next_epoch = spec.get_current_epoch(state) + 1
+    transition_to(spec, state, spec.compute_start_slot_at_epoch(next_epoch) - 2)
+
+    target_block = build_empty_block_for_next_slot(spec, state)
+    signed_target_block = state_transition_and_sign_block(spec, state, target_block)
+
+    # add target block to store
+    spec.on_block(store, signed_target_block)
+
+    # target checkpoint state is not yet in store
+
+    attestation_slot = target_block.slot + 1
+    transition_to(spec, state, attestation_slot)
+    attestation = get_valid_attestation(spec, state, slot=attestation_slot, signed=True)
+    assert attestation.data.target.root == target_block.hash_tree_root()
+
+    run_on_attestation(spec, state, store, attestation)
+
+
 @with_all_phases
 @spec_state_test
 def test_on_attestation_beacon_block_not_in_store(spec, state):
     store = spec.get_forkchoice_store(state)
-    time = store.time + spec.SECONDS_PER_SLOT * spec.SLOTS_PER_EPOCH
+    time = store.time + spec.SECONDS_PER_SLOT * (spec.SLOTS_PER_EPOCH + 1)
     spec.on_tick(store, time)
 
     # move to immediately before next epoch to make block new target
diff --git a/tests/core/pyspec/eth2spec/test/fork_choice/test_on_shard_head.py b/tests/core/pyspec/eth2spec/test/fork_choice/test_on_shard_head.py
index c84f073b2..a87a85917 100644
--- a/tests/core/pyspec/eth2spec/test/fork_choice/test_on_shard_head.py
+++ b/tests/core/pyspec/eth2spec/test/fork_choice/test_on_shard_head.py
@@ -22,7 +22,7 @@ def run_on_shard_block(spec, store, shard_store, signed_block, valid=True):
             assert False
 
     spec.on_shard_block(store, shard_store, signed_block)
-    assert shard_store.blocks[hash_tree_root(signed_block.message)] == signed_block.message
+    assert shard_store.signed_blocks[hash_tree_root(signed_block.message)] == signed_block
 
 
 def apply_shard_block(spec, store, shard_store, beacon_parent_state, shard_blocks_buffer):
@@ -41,10 +41,7 @@ def apply_shard_block(spec, store, shard_store, beacon_parent_state, shard_block
 
 
 def check_pending_shard_blocks(spec, store, shard_store, shard_blocks_buffer):
-    pending_shard_blocks = [
-        spec.SignedShardBlock(message=b)
-        for b in spec.get_pending_shard_blocks(store, shard_store)
-    ]
+    pending_shard_blocks = spec.get_pending_shard_blocks(store, shard_store)
     assert pending_shard_blocks == shard_blocks_buffer
 
 
@@ -72,7 +69,7 @@ def apply_shard_and_beacon(spec, state, store, shard_store, shard_blocks_buffer)
         shard_transitions = get_shard_transitions(
             spec,
             state,
-            shard_blocks={shard: shard_blocks_buffer},
+            shard_block_dict={shard: shard_blocks_buffer},
         )
         shard_transition = shard_transitions[shard]
         attestation = get_valid_on_time_attestation(
diff --git a/tests/core/pyspec/eth2spec/test/helpers/attestations.py b/tests/core/pyspec/eth2spec/test/helpers/attestations.py
index 0c3f012f5..3445f9ff6 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/attestations.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/attestations.py
@@ -83,14 +83,14 @@ def build_attestation_data(spec, state, slot, index, shard=None, shard_transitio
         attestation_data.shard = shard
 
         if shard_transition is not None:
-            lastest_shard_data_root_index = len(shard_transition.shard_data_roots) - 1
-            attestation_data.shard_head_root = shard_transition.shard_data_roots[lastest_shard_data_root_index]
+            last_offset_index = len(shard_transition.shard_data_roots) - 1
+            attestation_data.shard_head_root = shard_transition.shard_states[last_offset_index].latest_block_root
             attestation_data.shard_transition_root = shard_transition.hash_tree_root()
         else:
             if on_time:
                 shard_transition = spec.get_shard_transition(state, shard, shard_blocks=[])
-                lastest_shard_data_root_index = len(shard_transition.shard_data_roots) - 1
-                attestation_data.shard_head_root = shard_transition.shard_data_roots[lastest_shard_data_root_index]
+                last_offset_index = len(shard_transition.shard_data_roots) - 1
+                attestation_data.shard_head_root = shard_transition.shard_states[last_offset_index].latest_block_root
                 attestation_data.shard_transition_root = shard_transition.hash_tree_root()
             else:
                 attestation_data.shard_head_root = state.shard_states[shard].latest_block_root
diff --git a/tests/core/pyspec/eth2spec/test/helpers/custody.py b/tests/core/pyspec/eth2spec/test/helpers/custody.py
index f63a07099..d25b91a41 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/custody.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/custody.py
@@ -172,7 +172,7 @@ def get_sample_shard_transition(spec, start_slot, block_lengths):
         start_slot=start_slot,
         shard_block_lengths=block_lengths,
         shard_data_roots=b,
-        shard_states=[spec.Root() for x in block_lengths],
+        shard_states=[spec.ShardState() for x in block_lengths],
         proposer_signature_aggregate=spec.BLSSignature(),
     )
     return shard_transition
diff --git a/tests/core/pyspec/eth2spec/test/helpers/shard_block.py b/tests/core/pyspec/eth2spec/test/helpers/shard_block.py
index 6957f2283..c8d60938b 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/shard_block.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/shard_block.py
@@ -30,7 +30,7 @@ def build_shard_block(spec,
         slot = shard_parent_state.slot + 1
 
     if body is None:
-        body = b'\x56' * 128
+        body = get_sample_shard_block_body(spec)
 
     beacon_state, beacon_parent_root = get_state_and_beacon_parent_root_at_slot(spec, beacon_state, slot)
     proposer_index = spec.get_shard_proposer_index(beacon_state, slot, shard)
@@ -52,10 +52,10 @@ def build_shard_block(spec,
     return signed_block
 
 
-def get_shard_transitions(spec, parent_beacon_state, shard_blocks):
+def get_shard_transitions(spec, parent_beacon_state, shard_block_dict):
     shard_transitions = [spec.ShardTransition()] * spec.MAX_SHARDS
     on_time_slot = parent_beacon_state.slot + 1
-    for shard, blocks in shard_blocks.items():
+    for shard, blocks in shard_block_dict.items():
         shard_transition = spec.get_shard_transition(parent_beacon_state, shard, blocks)
         offset_slots = spec.compute_offset_slots(
             spec.get_latest_slot_for_shard(parent_beacon_state, shard),
@@ -81,3 +81,8 @@ def get_committee_index_of_shard(spec, state, slot, shard):  # Optional[Committe
         if (start_shard + committee_index) % active_shard_count == shard:
             return committee_index
     return None
+
+
+def get_sample_shard_block_body(spec, is_max=False):
+    size = spec.MAX_SHARD_BLOCK_SIZE if is_max else 128
+    return b'\x56' * size
diff --git a/tests/core/pyspec/eth2spec/test/phase0/block_processing/test_process_attester_slashing.py b/tests/core/pyspec/eth2spec/test/phase0/block_processing/test_process_attester_slashing.py
index 063514498..192e0390d 100644
--- a/tests/core/pyspec/eth2spec/test/phase0/block_processing/test_process_attester_slashing.py
+++ b/tests/core/pyspec/eth2spec/test/phase0/block_processing/test_process_attester_slashing.py
@@ -197,6 +197,71 @@ def test_participants_already_slashed(spec, state):
 # Some of the following tests are phase0 only: phase 1 lists participants with bitfields instead of index list.
 
 
+@with_phases([PHASE0])
+@spec_state_test
+@always_bls
+def test_att1_high_index(spec, state):
+    attester_slashing = get_valid_attester_slashing(spec, state, signed_1=True, signed_2=True)
+
+    indices = get_indexed_attestation_participants(spec, attester_slashing.attestation_1)
+    indices.append(spec.ValidatorIndex(len(state.validators)))  # off by 1
+    attester_slashing.attestation_1.attesting_indices = indices
+
+    yield from run_attester_slashing_processing(spec, state, attester_slashing, False)
+
+
+@with_phases([PHASE0])
+@spec_state_test
+@always_bls
+def test_att2_high_index(spec, state):
+    attester_slashing = get_valid_attester_slashing(spec, state, signed_1=True, signed_2=True)
+
+    indices = get_indexed_attestation_participants(spec, attester_slashing.attestation_2)
+    indices.append(spec.ValidatorIndex(len(state.validators)))  # off by 1
+    attester_slashing.attestation_2.attesting_indices = indices
+
+    yield from run_attester_slashing_processing(spec, state, attester_slashing, False)
+
+
+@with_phases([PHASE0])
+@spec_state_test
+@always_bls
+def test_att1_empty_indices(spec, state):
+    attester_slashing = get_valid_attester_slashing(spec, state, signed_1=False, signed_2=True)
+
+    attester_slashing.attestation_1.attesting_indices = []
+    attester_slashing.attestation_1.signature = spec.bls.Z2_SIGNATURE
+
+    yield from run_attester_slashing_processing(spec, state, attester_slashing, False)
+
+
+@with_phases([PHASE0])
+@spec_state_test
+@always_bls
+def test_att2_empty_indices(spec, state):
+    attester_slashing = get_valid_attester_slashing(spec, state, signed_1=True, signed_2=False)
+
+    attester_slashing.attestation_2.attesting_indices = []
+    attester_slashing.attestation_2.signature = spec.bls.Z2_SIGNATURE
+
+    yield from run_attester_slashing_processing(spec, state, attester_slashing, False)
+
+
+@with_phases([PHASE0])
+@spec_state_test
+@always_bls
+def test_all_empty_indices(spec, state):
+    attester_slashing = get_valid_attester_slashing(spec, state, signed_1=False, signed_2=False)
+
+    attester_slashing.attestation_1.attesting_indices = []
+    attester_slashing.attestation_1.signature = spec.bls.Z2_SIGNATURE
+
+    attester_slashing.attestation_2.attesting_indices = []
+    attester_slashing.attestation_2.signature = spec.bls.Z2_SIGNATURE
+
+    yield from run_attester_slashing_processing(spec, state, attester_slashing, False)
+
+
 @with_phases([PHASE0])
 @spec_state_test
 @always_bls
diff --git a/tests/core/pyspec/eth2spec/test/phase1/block_processing/test_process_shard_transition.py b/tests/core/pyspec/eth2spec/test/phase1/block_processing/test_process_shard_transition.py
index 6d8878177..b14b836c0 100644
--- a/tests/core/pyspec/eth2spec/test/phase1/block_processing/test_process_shard_transition.py
+++ b/tests/core/pyspec/eth2spec/test/phase1/block_processing/test_process_shard_transition.py
@@ -3,69 +3,208 @@ from eth2spec.test.context import (
     with_all_phases_except,
     spec_state_test,
 )
-from eth2spec.test.helpers.attestations import get_valid_on_time_attestation
+from eth2spec.test.helpers.attestations import (
+    get_valid_attestation,
+    get_valid_on_time_attestation,
+    run_attestation_processing,
+)
 from eth2spec.test.helpers.shard_transitions import run_shard_transitions_processing
 from eth2spec.test.helpers.shard_block import (
     build_shard_block,
     get_shard_transitions,
+    get_sample_shard_block_body,
+    get_committee_index_of_shard,
 )
 from eth2spec.test.helpers.state import transition_to, transition_to_valid_shard_slot, next_slot
 
 
-def run_basic_crosslink_tests(spec, state, target_len_offset_slot, valid=True):
+def get_initial_env(spec, state, target_len_offset_slot):
     state = transition_to_valid_shard_slot(spec, state)
     committee_index = spec.CommitteeIndex(0)
+    target_shard_slot = state.slot + target_len_offset_slot - 1
+    shard = spec.compute_shard_from_committee_index(state, committee_index, target_shard_slot)
+    assert state.shard_states[shard].slot == state.slot - 1
+    return state, shard, target_shard_slot
+
+
+def get_attestations_and_shard_transitions(spec, state, shard_block_dict):
+    shard_transitions = get_shard_transitions(spec, state, shard_block_dict)
+    attestations = [
+        get_valid_on_time_attestation(
+            spec, state,
+            index=get_committee_index_of_shard(spec, state, state.slot, shard),
+            shard_transition=shard_transition,
+            signed=True,
+        )
+        for shard, shard_transition in enumerate(shard_transitions)
+        if shard_transition != spec.ShardTransition()
+    ]
+    return attestations, shard_transitions
+
+
+def is_full_crosslink(spec, state):
+    epoch = spec.compute_epoch_at_slot(state.slot)
+    return spec.get_committee_count_per_slot(state, epoch) >= spec.get_active_shard_count(state)
+
+
+def run_successful_crosslink_tests(spec, state, target_len_offset_slot):
+    state, shard, target_shard_slot = get_initial_env(spec, state, target_len_offset_slot)
     init_slot = state.slot
-    shard_slot = state.slot + target_len_offset_slot - 1
-    shard = spec.compute_shard_from_committee_index(state, committee_index, shard_slot)
-    assert state.shard_states[shard].slot == init_slot - 1
 
-    # Create SignedShardBlock
-    body = b'\x56' * spec.MAX_SHARD_BLOCK_SIZE
-    shard_block = build_shard_block(spec, state, shard, body=body, slot=state.slot, signed=True)
-    shard_blocks = [shard_block]
+    # Create SignedShardBlock at init_slot
+    shard_block = build_shard_block(
+        spec, state, shard,
+        slot=init_slot, body=get_sample_shard_block_body(spec, is_max=True), signed=True
+    )
+
+    # Transition state to target shard slot
+    transition_to(spec, state, target_shard_slot)
+
+    # Create a shard_transitions that would be included at beacon block `target_shard_slot + 1`
+    shard_block_dict = {shard: [shard_block]}
+    attestations, shard_transitions = get_attestations_and_shard_transitions(spec, state, shard_block_dict)
 
-    # Transition state latest shard slot
-    transition_to(spec, state, shard_slot)
-    # Create a shard_transitions that would be included at beacon block `state.slot + target_len_offset_slot`
-    shard_transitions = get_shard_transitions(
-        spec,
-        state,
-        shard_blocks={shard: shard_blocks},
-    )
-    shard_transition = shard_transitions[shard]
-    attestation = get_valid_on_time_attestation(
-        spec,
-        state,
-        index=committee_index,
-        shard_transition=shard_transition,
-        signed=False,
-    )
     next_slot(spec, state)
+
+    for attestation in attestations:
+        _, _, _ = run_attestation_processing(spec, state, attestation)
+
+    _, winning_roots = spec.get_shard_winning_roots(state, attestations)
+    assert len(winning_roots) == 1
+    shard_transition = shard_transitions[shard]
+    assert winning_roots[0] == shard_transition.hash_tree_root()
+
     pre_gasprice = state.shard_states[shard].gasprice
+    pre_shard_states = state.shard_states.copy()
+    yield from run_shard_transitions_processing(spec, state, shard_transitions, attestations)
 
-    transition_to(spec, state, init_slot + target_len_offset_slot)
-    pre_shard_state = state.shard_states[shard]
-    yield from run_shard_transitions_processing(spec, state, shard_transitions, [attestation], valid=valid)
+    for index, shard_state in enumerate(state.shard_states):
+        if index == shard:
+            assert shard_state != pre_shard_states[index]
+            assert shard_state == shard_transition.shard_states[len(shard_transition.shard_states) - 1]
+            assert shard_state.latest_block_root == shard_block.message.hash_tree_root()
+            if target_len_offset_slot == 1:
+                assert shard_state.gasprice > pre_gasprice
+        else:
+            assert shard_state == pre_shard_states[index]
 
-    if valid:
-        shard_state = state.shard_states[shard]
-        assert shard_state != pre_shard_state
-        assert shard_state == shard_transition.shard_states[len(shard_transition.shard_states) - 1]
-        assert shard_state.latest_block_root == shard_block.message.hash_tree_root()
-        if target_len_offset_slot == 1:
-            assert shard_state.gasprice > pre_gasprice
+    for pending_attestation in state.current_epoch_attestations:
+        assert bool(pending_attestation.crosslink_success) is True
 
 
 @with_all_phases_except([PHASE0])
 @spec_state_test
 def test_basic_crosslinks(spec, state):
-    # NOTE: this test is only for full crosslink (minimal config), not for mainnet
-    yield from run_basic_crosslink_tests(spec, state, target_len_offset_slot=1, valid=True)
+    if not is_full_crosslink(spec, state):
+        # Skip this test
+        return
+
+    yield from run_successful_crosslink_tests(spec, state, target_len_offset_slot=1)
 
 
 @with_all_phases_except([PHASE0])
 @spec_state_test
 def test_multiple_offset_slots(spec, state):
-    # NOTE: this test is only for full crosslink (minimal config), not for mainnet
-    yield from run_basic_crosslink_tests(spec, state, target_len_offset_slot=2, valid=True)
+    if not is_full_crosslink(spec, state):
+        # Skip this test
+        return
+
+    yield from run_successful_crosslink_tests(spec, state, target_len_offset_slot=2)
+
+
+@with_all_phases_except([PHASE0])
+@spec_state_test
+def test_no_winning_root(spec, state):
+    if not is_full_crosslink(spec, state):
+        # Skip this test
+        return
+
+    state, shard, target_shard_slot = get_initial_env(spec, state, target_len_offset_slot=1)
+    init_slot = state.slot
+
+    # Create SignedShardBlock at init_slot
+    shard_block = build_shard_block(
+        spec, state, shard,
+        slot=init_slot, body=get_sample_shard_block_body(spec, is_max=True), signed=True
+    )
+
+    # Transition state to target shard slot
+    transition_to(spec, state, target_shard_slot)
+
+    # Create a shard_transitions that would be included at beacon block `target_shard_slot + 1`
+    shard_transitions = get_shard_transitions(spec, state, {shard: [shard_block]})
+    shard_transition = shard_transitions[shard]
+    committee_index = get_committee_index_of_shard(spec, state, state.slot, shard)
+    attestation = get_valid_attestation(
+        spec, state,
+        index=committee_index,
+        shard_transition=shard_transition,
+        # Decrease attested participants to 1/3 committee
+        filter_participant_set=lambda committee: set(list(committee)[:len(committee) // 3]),
+        signed=True,
+        on_time=True,
+    )
+
+    next_slot(spec, state)
+
+    _, _, _ = run_attestation_processing(spec, state, attestation)
+
+    _, winning_roots = spec.get_shard_winning_roots(state, [attestation])
+    assert len(winning_roots) == 0
+
+    # No winning root, shard_transitions[shard] is empty
+    shard_transitions = [spec.ShardTransition()] * spec.MAX_SHARDS
+    pre_shard_states = state.shard_states.copy()
+    yield from run_shard_transitions_processing(spec, state, shard_transitions, [attestation])
+
+    for pending_attestation in state.current_epoch_attestations:
+        assert bool(pending_attestation.crosslink_success) is False
+
+    assert state.shard_states == pre_shard_states
+
+
+@with_all_phases_except([PHASE0])
+@spec_state_test
+def test_wrong_shard_transition_root(spec, state):
+    if not is_full_crosslink(spec, state):
+        # Skip this test
+        return
+
+    state, shard, target_shard_slot = get_initial_env(spec, state, target_len_offset_slot=1)
+    init_slot = state.slot
+
+    # Create SignedShardBlock at init_slot
+    shard_block = build_shard_block(
+        spec, state, shard,
+        slot=init_slot, body=get_sample_shard_block_body(spec, is_max=True), signed=True
+    )
+
+    # Transition state to target shard slot
+    transition_to(spec, state, target_shard_slot)
+
+    # Create a shard_transitions that would be included at beacon block `target_shard_slot + 1`
+    shard_transitions = get_shard_transitions(spec, state, {shard: [shard_block]})
+    shard_transition = shard_transitions[shard]
+    wrong_shard_transition = shard_transition.copy()
+    wrong_shard_transition.shard_states[shard].gasprice = shard_transition.shard_states[shard].gasprice + 1
+    committee_index = get_committee_index_of_shard(spec, state, state.slot, shard)
+    attestation = get_valid_attestation(
+        spec, state,
+        index=committee_index,
+        shard_transition=wrong_shard_transition,
+        signed=True,
+        on_time=True,
+    )
+    attestations = [attestation]
+
+    next_slot(spec, state)
+
+    run_attestation_processing(spec, state, attestation)
+
+    # Check if winning root != shard_transition.hash_tree_root()
+    _, winning_roots = spec.get_shard_winning_roots(state, attestations)
+    assert len(winning_roots) == 1
+    shard_transition = shard_transitions[shard]
+    assert winning_roots[0] != shard_transition.hash_tree_root()
+
+    yield from run_shard_transitions_processing(spec, state, shard_transitions, attestations, valid=False)
diff --git a/tests/core/pyspec/eth2spec/test/phase1/sanity/test_blocks.py b/tests/core/pyspec/eth2spec/test/phase1/sanity/test_blocks.py
index 33b0beac7..aa24bfc1e 100644
--- a/tests/core/pyspec/eth2spec/test/phase1/sanity/test_blocks.py
+++ b/tests/core/pyspec/eth2spec/test/phase1/sanity/test_blocks.py
@@ -19,9 +19,9 @@ def run_beacon_block_with_shard_blocks(spec, state, target_len_offset_slot, comm
 
     body = b'\x56' * spec.MAX_SHARD_BLOCK_SIZE
     shard_block = build_shard_block(spec, state, shard, body=body, slot=state.slot, signed=True)
-    shard_blocks: Dict[spec.Shard, Sequence[spec.SignedShardBlock]] = {shard: [shard_block]}
+    shard_block_dict: Dict[spec.Shard, Sequence[spec.SignedShardBlock]] = {shard: [shard_block]}
 
-    shard_transitions = get_shard_transitions(spec, state, shard_blocks)
+    shard_transitions = get_shard_transitions(spec, state, shard_block_dict)
     attestations = [
         get_valid_on_time_attestation(
             spec,
@@ -30,7 +30,7 @@ def run_beacon_block_with_shard_blocks(spec, state, target_len_offset_slot, comm
             shard_transition=shard_transitions[shard],
             signed=True,
         )
-        for shard in shard_blocks.keys()
+        for shard in shard_block_dict.keys()
     ]
 
     beacon_block = build_empty_block(spec, state, slot=state.slot + 1)
@@ -50,16 +50,16 @@ def run_beacon_block_with_shard_blocks(spec, state, target_len_offset_slot, comm
 
     for shard in range(spec.get_active_shard_count(state)):
         post_shard_state = state.shard_states[shard]
-        if shard in shard_blocks:
+        if shard in shard_block_dict:
             # Shard state has been changed to state_transition result
             assert post_shard_state == shard_transitions[shard].shard_states[
                 len(shard_transitions[shard].shard_states) - 1
             ]
             assert post_shard_state.slot == state.slot - 1
-            if len(shard_blocks[shard]) == 0:
+            if len((shard_block_dict[shard])) == 0:
                 # `latest_block_root` is the same
                 assert post_shard_state.latest_block_root == pre_shard_states[shard].latest_block_root
-            if target_len_offset_slot == 1 and len(shard_blocks) > 0:
+            if target_len_offset_slot == 1 and len(shard_block_dict[shard]) > 0:
                 assert post_shard_state.gasprice > pre_gasprice