diff --git a/gpbft/chain.go b/gpbft/chain.go
index 83a75164..a759e481 100644
--- a/gpbft/chain.go
+++ b/gpbft/chain.go
@@ -288,7 +288,7 @@ func (c ECChain) Validate() error {
 // Returns an identifier for the chain suitable for use as a map key.
 // This must completely determine the sequence of tipsets in the chain.
 func (c ECChain) Key() ChainKey {
-	ln := len(c) * (8 + 32 + 4) // epoch + commitement + ts length
+	ln := len(c) * (8 + 32 + 4) // epoch + commitment + ts length
 	for i := range c {
 		ln += len(c[i].Key) + len(c[i].PowerTable)
 	}
diff --git a/gpbft/gpbft.go b/gpbft/gpbft.go
index 20a1e695..7c0b53fb 100644
--- a/gpbft/gpbft.go
+++ b/gpbft/gpbft.go
@@ -184,17 +184,23 @@ type instance struct {
 	// Supplemental data that all participants must agree on ahead of time. Messages that
 	// propose supplemental data that differs with our supplemental data will be discarded.
 	supplementalData *SupplementalData
+	// initialProposal represents the unmodified initial input chain of this
+	// instance. Constructed when the instance is initialized, the initialProposal is
+	// utilized to update candidates during the reception of late-arriving QUALITY
+	// messages to aid faster convergence.
+	initialProposal ECChain
 	// This instance's proposal for the current round. Never bottom.
 	// This is set after the QUALITY phase, and changes only at the end of a full round.
 	proposal ECChain
 	// The value to be transmitted at the next phase, which may be bottom.
 	// This value may change away from the proposal between phases.
 	value ECChain
-	// The set of values that are acceptable candidates to this instance.
-	// This includes the base chain, all prefixes of proposal that found a strong quorum
-	// of support in the QUALITY phase, and any chains that could possibly have been
-	// decided by another participant.
-	candidates []ECChain
+	// candidates contains a set of values that are acceptable candidates to this
+	// instance. This includes the base chain, all prefixes of proposal that found a
+	// strong quorum of support in the QUALITY phase or late arriving quality
+	// messages, including any chains that could possibly have been decided by
+	// another participant.
+	candidates map[ChainKey]struct{}
 	// The final termination value of the instance, for communication to the participant.
 	// This field is an alternative to plumbing an optional decision value out through
 	// all the method calls, or holding a callback handle to receive it here.
@@ -235,11 +241,14 @@ func newInstance(
 		round:            0,
 		phase:            INITIAL_PHASE,
 		supplementalData: data,
+		initialProposal:  input,
 		proposal:         input,
 		broadcasted:      newBroadcastState(),
 		value:            ECChain{},
-		candidates:       []ECChain{input.BaseChain()},
-		quality:          newQuorumState(powerTable),
+		candidates: map[ChainKey]struct{}{
+			input.BaseChain().Key(): {},
+		},
+		quality: newQuorumState(powerTable),
 		rounds: map[uint64]*roundState{
 			0: newRoundState(powerTable),
 		},
@@ -351,11 +360,9 @@ func (i *instance) receiveOne(msg *GMessage) (bool, error) {
 	if i.phase == TERMINATED_PHASE {
 		return false, nil // No-op
 	}
-	// Ignore QUALITY messages after exiting the QUALITY phase.
 	// Ignore CONVERGE and PREPARE messages for prior rounds.
 	forPriorRound := msg.Vote.Round < i.round
-	if (msg.Vote.Step == QUALITY_PHASE && i.phase != QUALITY_PHASE) ||
-		(forPriorRound && msg.Vote.Step == CONVERGE_PHASE) ||
+	if (forPriorRound && msg.Vote.Step == CONVERGE_PHASE) ||
 		(forPriorRound && msg.Vote.Step == PREPARE_PHASE) {
 		return false, nil
 	}
@@ -373,7 +380,8 @@ func (i *instance) receiveOne(msg *GMessage) (bool, error) {
 	msgRound := i.getRound(msg.Vote.Round)
 	switch msg.Vote.Step {
 	case QUALITY_PHASE:
-		// Receive each prefix of the proposal independently.
+		// Receive each prefix of the proposal independently, which is accepted at any
+		// round/phase.
 		i.quality.ReceiveEachPrefix(msg.Sender, msg.Vote.Value)
 	case CONVERGE_PHASE:
 		if err := msgRound.converged.Receive(msg.Sender, i.powerTable, msg.Vote.Value, msg.Ticket, msg.Justification); err != nil {
@@ -498,12 +506,8 @@ func (i *instance) tryQuality() error {
 	}
 
 	if foundQuorum || timeoutExpired {
-		// Add prefixes with quorum to candidates (skipping base chain, which is already there).
-		for l := range i.proposal {
-			if l > 0 {
-				i.candidates = append(i.candidates, i.proposal.Prefix(l))
-			}
-		}
+		// Add prefixes with quorum to candidates.
+		i.addCandidatePrefixes(i.proposal)
 		i.value = i.proposal
 		i.log("adopting proposal/value %s", &i.proposal)
 		i.beginPrepare(nil)
@@ -544,6 +548,17 @@ func (i *instance) tryConverge() error {
 		return nil
 	}
 
+	// Find the first initial proposal prefix that has strong quorum as a result of
+	// arriving QUALITY messages and update candidates for its consecutive prefixes.
+	for p := range i.initialProposal {
+		initialProposalPrefix := i.initialProposal.Prefix(p)
+		if foundQuorum := i.quality.HasStrongQuorumFor(initialProposalPrefix.Key()); foundQuorum {
+			i.log("expanding candidates for proposal %s due to late QUALITY quorum for %s", i.proposal, &initialProposalPrefix)
+			i.addCandidatePrefixes(initialProposalPrefix)
+			break
+		}
+	}
+
 	winner := i.getRound(i.round).converged.FindMaxTicketProposal(i.powerTable)
 	if !winner.IsValid() {
 		return fmt.Errorf("no values at CONVERGE")
@@ -555,11 +570,11 @@ func (i *instance) tryConverge() error {
 	// in the last round, consider it a candidate.
 	if !i.isCandidate(winner.Chain) && winner.Justification.Vote.Step == PREPARE_PHASE && possibleDecisionLastRound {
 		i.log("⚠️ swaying from %s to %s by CONVERGE", &i.proposal, &winner.Chain)
-		i.candidates = append(i.candidates, winner.Chain)
+		i.addCandidate(winner.Chain)
 	}
 	if i.isCandidate(winner.Chain) {
 		i.proposal = winner.Chain
-		i.log("adopting proposal %s after converge", &winner.Chain)
+		i.log("adopting proposal %s after CONVERGE", &winner.Chain)
 	} else {
 		// Else preserve own proposal.
 		// This could alternatively loop to next lowest ticket as an optimisation to increase the
@@ -648,7 +663,8 @@ func (i *instance) tryCommit(round uint64) error {
 	committed := i.getRound(round).committed
 	quorumValue, foundStrongQuorum := committed.FindStrongQuorumValue()
 	timedOut := atOrAfter(i.participant.host.Time(), i.phaseTimeout)
-	phaseComplete := timedOut && committed.ReceivedFromStrongQuorum()
+	receivedFromStrongQuorum := committed.ReceivedFromStrongQuorum()
+	phaseComplete := timedOut && receivedFromStrongQuorum
 
 	if foundStrongQuorum && !quorumValue.IsZero() {
 		// A participant may be forced to decide a value that's not its preferred chain.
@@ -668,7 +684,7 @@ func (i *instance) tryCommit(round uint64) error {
 				if !v.IsZero() {
 					if !i.isCandidate(v) {
 						i.log("⚠️ swaying from %s to %s by COMMIT", &i.input, &v)
-						i.candidates = append(i.candidates, v)
+						i.addCandidate(v)
 					}
 					if !v.Eq(i.proposal) {
 						i.proposal = v
@@ -786,7 +802,7 @@ func (i *instance) skipToRound(round uint64, chain ECChain, justification *Justi
 
 	if justification.Vote.Step == PREPARE_PHASE {
 		i.log("⚠️ swaying from %s to %s by skip to round %d", &i.proposal, chain, i.round)
-		i.candidates = append(i.candidates, chain)
+		i.addCandidate(chain)
 		i.proposal = chain
 	}
 	i.beginConverge(justification)
@@ -795,12 +811,21 @@ func (i *instance) skipToRound(round uint64, chain ECChain, justification *Justi
 // Returns whether a chain is acceptable as a proposal for this instance to vote for.
 // This is "EC Compatible" in the pseudocode.
 func (i *instance) isCandidate(c ECChain) bool {
-	for _, candidate := range i.candidates {
-		if c.Eq(candidate) {
-			return true
-		}
+	_, exists := i.candidates[c.Key()]
+	return exists
+}
+
+func (i *instance) addCandidatePrefixes(c ECChain) {
+	for l := range c {
+		i.addCandidate(c.Prefix(l))
+	}
+}
+
+func (i *instance) addCandidate(c ECChain) {
+	key := c.Key()
+	if _, exists := i.candidates[key]; !exists {
+		i.candidates[key] = struct{}{}
 	}
-	return false
 }
 
 func (i *instance) terminate(decision *Justification) {
@@ -1279,9 +1304,9 @@ func newConvergeState() *convergeState {
 	}
 }
 
-// SetSelfValue sets the participant's locally-proposed converge value.
-// This means the participant need not rely on messages broadcast to be received by itself.
-// See HasSelfValue.
+// SetSelfValue sets the participant's locally-proposed converge value. This
+// means the participant need not to rely on messages broadcast to be received by
+// itself.
 func (c *convergeState) SetSelfValue(value ECChain, justification *Justification) {
 	// any converge for the given value is better than self-reported
 	// as self-reported has no ticket