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We improve the fundamental security threshold of eventual consensusProof-of-Stake (PoS) blockchain protocols under the longest-chain rule byshowing, for the first time, the positive effect of rounds with concurrenthonest leaders.  Current security analyses reduce consistency to the dynamics of an abstract,round-based block creation process that is determined by three eventsassociated with a round: (i) event $A$: at least one adversarial leader, (ii)event $S$: a single honest leader, and (iii) event $M$: multiple, but honest,leaders. We present an asymptotically optimal consistency analysis assumingthat an honest round is more likely than an adversarial round (i.e., $\Pr[S] +\Pr[M] > \Pr[A]$); this threshold is optimal. This is a first in the literatureand can be applied to both the simple synchronous communication as well ascommunication with bounded delays.  In all existing consistency analyses, event $M$ is either penalized ortreated neutrally. Specifically, the consistency analyses in Ouroboros Praos(Eurocrypt 2018) and Genesis (CCS 2018) assume that $\Pr[S] - \Pr[M] > \Pr[A]$;the analyses in Sleepy Consensus (Asiacrypt 2017) and Snow White (Fin. Crypto2019) assume that $\Pr[S] > \Pr[A]$. Moreover, all existing analyses completelybreak down when $\Pr[S] < \Pr[A]$. These thresholds determine the criticaltrade-off between the honest majority, network delays, and consistency error.  Our new results can be directly applied to improve the security guarantees ofthe existing protocols. We also provide an efficient algorithm to explicitlycalculate these error probabilities in the synchronous setting. Furthermore, wecomplement these results by analyzing the setting where $S$ is rare, evenallowing $\Pr[S] = 0$, under the added assumption that honest players adopt aconsistent chain selection rule.

Consistency of Proof-of-Stake Blockchains with Concurrent Honest Slot Leaders