diff --git a/specs/phase0/beacon-chain.md b/specs/phase0/beacon-chain.md
index 5f04269e1..262cc6797 100644
--- a/specs/phase0/beacon-chain.md
+++ b/specs/phase0/beacon-chain.md
@@ -1225,7 +1225,7 @@ def verify_block_signature(state: BeaconState, signed_block: SignedBeaconBlock)
 
 ```python
 def process_slots(state: BeaconState, slot: Slot) -> None:
-    assert state.slot <= slot
+    assert state.slot < slot
     while state.slot < slot:
         process_slot(state)
         # Process epoch on the start slot of the next epoch
@@ -1584,6 +1584,8 @@ def process_block(state: BeaconState, block: BeaconBlock) -> None:
 def process_block_header(state: BeaconState, block: BeaconBlock) -> None:
     # Verify that the slots match
     assert block.slot == state.slot
+    # Verify that the block is newer than latest block header
+    assert block.slot > state.latest_block_header.slot
     # Verify that proposer index is the correct index
     assert block.proposer_index == get_beacon_proposer_index(state)
     # Verify that the parent matches
diff --git a/tests/core/pyspec/eth2spec/test/helpers/attestations.py b/tests/core/pyspec/eth2spec/test/helpers/attestations.py
index 1baed6989..e93a63c28 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/attestations.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/attestations.py
@@ -283,7 +283,8 @@ def fill_aggregate_attestation(spec, state, attestation, signed=False):
 
 
 def add_attestations_to_state(spec, state, attestations, slot):
-    spec.process_slots(state, slot)
+    if state.slot < slot:
+        spec.process_slots(state, slot)
     for attestation in attestations:
         spec.process_attestation(state, attestation)
 
diff --git a/tests/core/pyspec/eth2spec/test/helpers/block.py b/tests/core/pyspec/eth2spec/test/helpers/block.py
index e40d5e7d8..69cb59021 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/block.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/block.py
@@ -15,7 +15,8 @@ def get_proposer_index_maybe(spec, state, slot, proposer_index=None):
                       " Signing block is slow due to transition for proposer index calculation.")
             # use stub state to get proposer index of future slot
             stub_state = state.copy()
-            spec.process_slots(stub_state, slot)
+            if stub_state.slot < slot:
+                spec.process_slots(stub_state, slot)
             proposer_index = spec.get_beacon_proposer_index(stub_state)
     return proposer_index
 
@@ -52,15 +53,19 @@ def sign_block(spec, state, block, proposer_index=None):
 
 
 def transition_unsigned_block(spec, state, block):
-    spec.process_slots(state, block.slot)
+    assert state.slot < block.slot  # Preserve assertion from state transition to avoid strange pre-states from testing
+    if state.slot < block.slot:
+        spec.process_slots(state, block.slot)
+    assert state.latest_block_header.slot < block.slot  # There may not already be a block in this slot or past it.
+    assert state.slot == block.slot  # The block must be for this slot
     spec.process_block(state, block)
 
 
-def apply_empty_block(spec, state):
+def apply_empty_block(spec, state, slot=None):
     """
     Transition via an empty block (on current slot, assuming no block has been applied yet).
     """
-    block = build_empty_block(spec, state)
+    block = build_empty_block(spec, state, slot)
     transition_unsigned_block(spec, state, block)
 
 
@@ -71,6 +76,12 @@ def build_empty_block(spec, state, slot=None):
     """
     if slot is None:
         slot = state.slot
+    if slot < state.slot:
+        raise Exception("build_empty_block cannot build blocks for past slots")
+    if state.slot < slot:
+        # transition forward in copied state to grab relevant data from state
+        state = state.copy()
+        spec.process_slots(state, slot)
 
     state, parent_block_root = get_state_and_beacon_parent_root_at_slot(spec, state, slot)
     empty_block = spec.BeaconBlock()
diff --git a/tests/core/pyspec/eth2spec/test/helpers/state.py b/tests/core/pyspec/eth2spec/test/helpers/state.py
index f08f22602..c70a47742 100644
--- a/tests/core/pyspec/eth2spec/test/helpers/state.py
+++ b/tests/core/pyspec/eth2spec/test/helpers/state.py
@@ -1,5 +1,5 @@
 from eth2spec.test.context import expect_assertion_error
-from eth2spec.test.helpers.block import sign_block, transition_unsigned_block
+from eth2spec.test.helpers.block import apply_empty_block, sign_block, transition_unsigned_block
 
 
 def get_balance(state, index):
@@ -17,7 +17,8 @@ def next_slots(spec, state, slots):
     """
     Transition given slots forward.
     """
-    spec.process_slots(state, state.slot + slots)
+    if slots > 0:
+        spec.process_slots(state, state.slot + slots)
 
 
 def transition_to(spec, state, slot):
@@ -30,6 +31,15 @@ def transition_to(spec, state, slot):
     assert state.slot == slot
 
 
+def transition_to_slot_via_block(spec, state, slot):
+    """
+    Transition to ``slot`` via an empty block transition
+    """
+    assert state.slot < slot
+    apply_empty_block(spec, state, slot)
+    assert state.slot == slot
+
+
 def transition_to_valid_shard_slot(spec, state):
     """
     Transition to slot `spec.PHASE_1_GENESIS_SLOT + 1` and fork at `spec.PHASE_1_GENESIS_SLOT`.
@@ -45,7 +55,15 @@ def next_epoch(spec, state):
     Transition to the start slot of the next epoch
     """
     slot = state.slot + spec.SLOTS_PER_EPOCH - (state.slot % spec.SLOTS_PER_EPOCH)
-    spec.process_slots(state, slot)
+    if slot > state.slot:
+        spec.process_slots(state, slot)
+
+
+def next_epoch_via_block(spec, state):
+    """
+    Transition to the start slot of the next epoch via a full block transition
+    """
+    apply_empty_block(spec, state, state.slot + spec.SLOTS_PER_EPOCH - state.slot % spec.SLOTS_PER_EPOCH)
 
 
 def get_state_root(spec, state, slot) -> bytes:
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attestation.py b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attestation.py
index 8663391aa..8752d3747 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attestation.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attestation.py
@@ -15,10 +15,9 @@ from eth2spec.test.helpers.attestations import (
 from eth2spec.test.helpers.state import (
     next_slot,
     next_slots,
-    next_epoch,
-    transition_to,
+    next_epoch_via_block,
+    transition_to_slot_via_block,
 )
-from eth2spec.test.helpers.block import apply_empty_block
 from eth2spec.utils.ssz.ssz_typing import Bitlist
 
 
@@ -47,9 +46,7 @@ def test_success_multi_proposer_index_iterations(spec, state):
 @spec_state_test
 def test_success_previous_epoch(spec, state):
     attestation = get_valid_attestation(spec, state, signed=True, on_time=False)
-    transition_to(spec, state, spec.SLOTS_PER_EPOCH - 1)
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     yield from run_attestation_processing(spec, state, attestation)
 
@@ -79,8 +76,7 @@ def test_after_epoch_slots(spec, state):
     attestation = get_valid_attestation(spec, state, signed=True, on_time=False)
 
     # increment past latest inclusion slot
-    transition_to(spec, state, state.slot + spec.SLOTS_PER_EPOCH + 1)
-    apply_empty_block(spec, state)
+    transition_to_slot_via_block(spec, state, state.slot + spec.SLOTS_PER_EPOCH + 1)
 
     yield from run_attestation_processing(spec, state, attestation, False)
 
@@ -151,8 +147,8 @@ def test_invalid_index(spec, state):
 @with_all_phases
 @spec_state_test
 def test_mismatched_target_and_slot(spec, state):
-    next_epoch(spec, state)
-    next_epoch(spec, state)
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     attestation = get_valid_attestation(spec, state, on_time=False)
     attestation.data.slot = attestation.data.slot - spec.SLOTS_PER_EPOCH
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attester_slashing.py b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attester_slashing.py
index 48dc75fd9..11ead6033 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attester_slashing.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_attester_slashing.py
@@ -5,10 +5,9 @@ from eth2spec.test.context import (
 from eth2spec.test.helpers.attestations import sign_indexed_attestation
 from eth2spec.test.helpers.attester_slashings import get_valid_attester_slashing, \
     get_indexed_attestation_participants, get_attestation_2_data, get_attestation_1_data
-from eth2spec.test.helpers.block import apply_empty_block
 from eth2spec.test.helpers.state import (
     get_balance,
-    next_epoch,
+    next_epoch_via_block,
 )
 
 
@@ -91,8 +90,7 @@ def test_success_double(spec, state):
 @with_all_phases
 @spec_state_test
 def test_success_surround(spec, state):
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     state.current_justified_checkpoint.epoch += 1
     attester_slashing = get_valid_attester_slashing(spec, state, signed_1=False, signed_2=True)
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_block_header.py b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_block_header.py
index a2eb744b9..b57090568 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_block_header.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/block_processing/test_process_block_header.py
@@ -9,7 +9,7 @@ def prepare_state_for_header_processing(spec, state):
     spec.process_slots(state, state.slot + 1)
 
 
-def run_block_header_processing(spec, state, block, valid=True):
+def run_block_header_processing(spec, state, block, prepare_state=True, valid=True):
     """
     Run ``process_block_header``, yielding:
       - pre-state ('pre')
@@ -17,7 +17,8 @@ def run_block_header_processing(spec, state, block, valid=True):
       - post-state ('post').
     If ``valid == False``, run expecting ``AssertionError``
     """
-    prepare_state_for_header_processing(spec, state)
+    if prepare_state:
+        prepare_state_for_header_processing(spec, state)
 
     yield 'pre', state
     yield 'block', block
@@ -68,6 +69,22 @@ def test_invalid_parent_root(spec, state):
     yield from run_block_header_processing(spec, state, block, valid=False)
 
 
+@with_all_phases
+@spec_state_test
+def test_invalid_multiple_blocks_single_slot(spec, state):
+    block = build_empty_block_for_next_slot(spec, state)
+
+    prepare_state_for_header_processing(spec, state)
+    spec.process_block_header(state, block)
+
+    assert state.latest_block_header.slot == state.slot
+
+    child_block = block.copy()
+    child_block.parent_root = block.hash_tree_root()
+
+    yield from run_block_header_processing(spec, state, child_block, prepare_state=False, valid=False)
+
+
 @with_all_phases
 @spec_state_test
 def test_proposer_slashed(spec, state):
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/epoch_processing/run_epoch_process_base.py b/tests/core/pyspec/eth2spec/test/phase_0/epoch_processing/run_epoch_process_base.py
index af4587a2a..b8692227f 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/epoch_processing/run_epoch_process_base.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/epoch_processing/run_epoch_process_base.py
@@ -18,7 +18,8 @@ def run_epoch_processing_to(spec, state, process_name: str):
     slot = state.slot + (spec.SLOTS_PER_EPOCH - state.slot % spec.SLOTS_PER_EPOCH)
 
     # transition state to slot before epoch state transition
-    spec.process_slots(state, slot - 1)
+    if state.slot < slot - 1:
+        spec.process_slots(state, slot - 1)
 
     # start transitioning, do one slot update before the epoch itself.
     spec.process_slot(state)
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_blocks.py b/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_blocks.py
index eebfe5171..a289d3b64 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_blocks.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_blocks.py
@@ -1,8 +1,14 @@
 from eth2spec.utils import bls
 
-from eth2spec.test.helpers.state import get_balance, state_transition_and_sign_block, next_slot, next_epoch
-from eth2spec.test.helpers.block import build_empty_block_for_next_slot, build_empty_block, sign_block, \
-    transition_unsigned_block
+from eth2spec.test.helpers.state import (
+    get_balance, state_transition_and_sign_block,
+    next_slot, next_epoch, next_epoch_via_block,
+)
+from eth2spec.test.helpers.block import (
+    build_empty_block_for_next_slot, build_empty_block,
+    sign_block,
+    transition_unsigned_block,
+)
 from eth2spec.test.helpers.keys import privkeys, pubkeys
 from eth2spec.test.helpers.attester_slashings import get_valid_attester_slashing, get_indexed_attestation_participants
 from eth2spec.test.helpers.proposer_slashings import get_valid_proposer_slashing, check_proposer_slashing_effect
@@ -46,10 +52,12 @@ def test_same_slot_block_transition(spec, state):
 
     yield 'pre', state
 
-    signed_block = state_transition_and_sign_block(spec, state, block)
+    assert state.slot == block.slot
+
+    signed_block = state_transition_and_sign_block(spec, state, block, expect_fail=True)
 
     yield 'blocks', [signed_block]
-    yield 'post', state
+    yield 'post', None
 
 
 @with_all_phases
@@ -72,6 +80,81 @@ def test_empty_block_transition(spec, state):
     assert spec.get_randao_mix(state, spec.get_current_epoch(state)) != spec.Bytes32()
 
 
+def process_and_sign_block_without_header_validations(spec, state, block):
+    """
+    Artificially bypass the restrictions in the state transition to transition and sign block
+
+    WARNING UNSAFE: Only use when generating valid-looking invalid blocks for test vectors
+    """
+
+    # Perform single mutation in `process_block_header`
+    state.latest_block_header = spec.BeaconBlockHeader(
+        slot=block.slot,
+        proposer_index=block.proposer_index,
+        parent_root=block.parent_root,
+        state_root=spec.Bytes32(),
+        body_root=block.body.hash_tree_root(),
+    )
+
+    # Perform rest of process_block transitions
+    spec.process_randao(state, block.body)
+    spec.process_eth1_data(state, block.body)
+    spec.process_operations(state, block.body)
+
+    # Insert post-state rot
+    block.state_root = state.hash_tree_root()
+
+    # Sign block
+    return sign_block(spec, state, block)
+
+
+@with_phases(['phase0'])
+@spec_state_test
+def test_proposal_for_genesis_slot(spec, state):
+    assert state.slot == spec.GENESIS_SLOT
+
+    yield 'pre', state
+
+    block = build_empty_block(spec, state, spec.GENESIS_SLOT)
+    block.parent_root = state.latest_block_header.hash_tree_root()
+
+    # Show that normal path through transition fails
+    failed_state = state.copy()
+    expect_assertion_error(
+        lambda: spec.state_transition(failed_state, spec.SignedBeaconBlock(message=block), validate_result=False)
+    )
+
+    # Artificially bypass the restriction in the state transition to transition and sign block for test vectors
+    signed_block = process_and_sign_block_without_header_validations(spec, state, block)
+
+    yield 'blocks', [signed_block]
+    yield 'post', None
+
+
+@with_all_phases
+@spec_state_test
+def test_parent_from_same_slot(spec, state):
+    yield 'pre', state
+
+    parent_block = build_empty_block_for_next_slot(spec, state)
+    signed_parent_block = state_transition_and_sign_block(spec, state, parent_block)
+
+    child_block = parent_block.copy()
+    child_block.parent_root = state.latest_block_header.hash_tree_root()
+
+    # Show that normal path through transition fails
+    failed_state = state.copy()
+    expect_assertion_error(
+        lambda: spec.state_transition(failed_state, spec.SignedBeaconBlock(message=child_block), validate_result=False)
+    )
+
+    # Artificially bypass the restriction in the state transition to transition and sign block for test vectors
+    signed_child_block = process_and_sign_block_without_header_validations(spec, state, child_block)
+
+    yield 'blocks', [signed_parent_block, signed_child_block]
+    yield 'post', None
+
+
 @with_all_phases
 @spec_state_test
 def test_invalid_state_root(spec, state):
@@ -378,22 +461,19 @@ def test_proposer_after_inactive_index(spec, state):
     state.validators[inactive_index].exit_epoch = spec.get_current_epoch(state)
 
     # skip forward, get brand new proposers
-    next_epoch(spec, state)
-    next_epoch(spec, state)
-    block = build_empty_block_for_next_slot(spec, state)
-    state_transition_and_sign_block(spec, state, block)
-
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
     while True:
-        next_slot(spec, state)
         proposer_index = spec.get_beacon_proposer_index(state)
         if proposer_index > inactive_index:
             # found a proposer that has a higher index than a disabled validator
             yield 'pre', state
             # test if the proposer can be recognized correctly after the inactive validator
-            signed_block = state_transition_and_sign_block(spec, state, build_empty_block(spec, state))
+            signed_block = state_transition_and_sign_block(spec, state, build_empty_block_for_next_slot(spec, state))
             yield 'blocks', [signed_block]
             yield 'post', state
             break
+        next_slot(spec, state)
 
 
 @with_all_phases
@@ -411,16 +491,16 @@ def test_high_proposer_index(spec, state):
 
     active_count = len(spec.get_active_validator_indices(state, current_epoch))
     while True:
-        next_slot(spec, state)
         proposer_index = spec.get_beacon_proposer_index(state)
         if proposer_index >= active_count:
             # found a proposer that has a higher index than the active validator count
             yield 'pre', state
             # test if the proposer can be recognized correctly, even while it has a high index.
-            signed_block = state_transition_and_sign_block(spec, state, build_empty_block(spec, state))
+            signed_block = state_transition_and_sign_block(spec, state, build_empty_block_for_next_slot(spec, state))
             yield 'blocks', [signed_block]
             yield 'post', state
             break
+        next_slot(spec, state)
 
 
 @with_all_phases
diff --git a/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_slots.py b/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_slots.py
index 6ef6be4d3..dd4f302ae 100644
--- a/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_slots.py
+++ b/tests/core/pyspec/eth2spec/test/phase_0/sanity/test_slots.py
@@ -51,7 +51,8 @@ def test_double_empty_epoch(spec, state):
 @with_all_phases
 @spec_state_test
 def test_over_epoch_boundary(spec, state):
-    spec.process_slots(state, state.slot + (spec.SLOTS_PER_EPOCH // 2))
+    if spec.SLOTS_PER_EPOCH > 1:
+        spec.process_slots(state, state.slot + (spec.SLOTS_PER_EPOCH // 2))
     yield 'pre', state
     slots = spec.SLOTS_PER_EPOCH
     yield 'slots', slots
diff --git a/tests/core/pyspec/eth2spec/test/phase_1/block_processing/test_process_early_derived_secret_reveal.py b/tests/core/pyspec/eth2spec/test/phase_1/block_processing/test_process_early_derived_secret_reveal.py
index 83b0fe325..668561261 100644
--- a/tests/core/pyspec/eth2spec/test/phase_1/block_processing/test_process_early_derived_secret_reveal.py
+++ b/tests/core/pyspec/eth2spec/test/phase_1/block_processing/test_process_early_derived_secret_reveal.py
@@ -1,6 +1,5 @@
 from eth2spec.test.helpers.custody import get_valid_early_derived_secret_reveal
-from eth2spec.test.helpers.block import apply_empty_block
-from eth2spec.test.helpers.state import next_epoch, get_balance
+from eth2spec.test.helpers.state import next_epoch_via_block, get_balance
 from eth2spec.test.context import (
     PHASE0,
     with_all_phases_except,
@@ -64,8 +63,7 @@ def test_reveal_from_current_epoch(spec, state):
 @spec_state_test
 @never_bls
 def test_reveal_from_past_epoch(spec, state):
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
     randao_key_reveal = get_valid_early_derived_secret_reveal(spec, state, spec.get_current_epoch(state) - 1)
 
     yield from run_early_derived_secret_reveal_processing(spec, state, randao_key_reveal, False)
diff --git a/tests/core/pyspec/eth2spec/test/test_finality.py b/tests/core/pyspec/eth2spec/test/test_finality.py
index 0b99cc3f9..55ccde2d9 100644
--- a/tests/core/pyspec/eth2spec/test/test_finality.py
+++ b/tests/core/pyspec/eth2spec/test/test_finality.py
@@ -1,7 +1,6 @@
 from eth2spec.test.context import spec_state_test, never_bls, with_all_phases, with_phases
-from eth2spec.test.helpers.state import next_epoch
+from eth2spec.test.helpers.state import next_epoch_via_block
 from eth2spec.test.helpers.attestations import next_epoch_with_attestations
-from eth2spec.test.helpers.block import apply_empty_block
 
 
 def check_finality(spec,
@@ -58,10 +57,8 @@ def test_finality_no_updates_at_genesis(spec, state):
 @never_bls
 def test_finality_rule_4(spec, state):
     # get past first two epochs that finality does not run on
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     yield 'pre', state
 
@@ -86,10 +83,8 @@ def test_finality_rule_4(spec, state):
 @never_bls
 def test_finality_rule_1(spec, state):
     # get past first two epochs that finality does not run on
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     yield 'pre', state
 
@@ -116,10 +111,8 @@ def test_finality_rule_1(spec, state):
 @never_bls
 def test_finality_rule_2(spec, state):
     # get past first two epochs that finality does not run on
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     yield 'pre', state
 
@@ -152,10 +145,8 @@ def test_finality_rule_3(spec, state):
     https://github.com/ethereum/eth2.0-specs/issues/611#issuecomment-463612892
     """
     # get past first two epochs that finality does not run on
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
-    next_epoch(spec, state)
-    apply_empty_block(spec, state)
+    next_epoch_via_block(spec, state)
+    next_epoch_via_block(spec, state)
 
     yield 'pre', state