The security of many Proof-of-Stake (PoS) payment systems relies on quorum-based State Machine Replication (SMR) protocols. While classical analyses assume purely Byzantine faults, real-world systems must tolerate both arbitrary failures and strategic, profit-driven validators. We therefore study quorum-based SMR under a hybrid model with honest, Byzantine, and rational participants. We first establish the fundamental limitations of traditional consensus mechanisms, proving two impossibility results: (1) in partially synchronous networks, no quorum-based protocol can achieve SMR when rational and Byzantine validators collectively exceed $1/3$ of the participants; and (2) even under synchronous network assumptions, SMR remains unattainable if this coalition comprises more than $2/3$ of the validator set. Assuming a synchrony bound $Δ$, we show how to extend any quorum-based SMR protocol to tolerate up to $1/3$ Byzantine and $1/3$ rational validators by modifying only its finalization rule. Our approach enforces a necessary bound on the total transaction volume finalized within any time window $Δ$ and introduces the \emph{strongest chain rule}, which enables efficient finalization of transactions when a supermajority of honest participants provably supports execution. Empirical analysis of Ethereum and Cosmos demonstrates validator participation exceeding the required $5/6$ threshold in over $99%$ of blocks, supporting the practicality of our design. Finally, we present a recovery mechanism that restores safety and liveness after consistency violations, even with up to $5/9$ Byzantine stake and $1/9$ rational stake, guaranteeing full reimbursement of provable client losses.

翻译：许多权益证明（PoS）支付系统的安全性依赖于基于法定人数的状态机复制（SMR）协议。经典分析假设的是纯粹的拜占庭故障，而现实世界的系统必须同时容忍任意故障和策略性的、以利润为导向的验证者。因此，我们在一个包含诚实、拜占庭和理性参与者的混合模型下研究基于法定人数的SMR。我们首先确立了传统共识机制的基本局限性，证明了两个不可能性结果：(1) 在部分同步网络中，当理性和拜占庭验证者合计超过参与者总数的 $1/3$ 时，任何基于法定人数的协议都无法实现SMR；(2) 即使在同步网络假设下，如果该联盟占验证者集合的比例超过 $2/3$，SMR仍然无法实现。假设同步边界为 $Δ$，我们展示了如何通过仅修改其最终确认规则，来扩展任何基于法定人数的SMR协议，以容忍高达 $1/3$ 的拜占庭验证者和 $1/3$ 的理性验证者。我们的方法强制规定了在任何时间窗口 $Δ$ 内最终确认的总交易量的必要上限，并引入了 \emph{最强链规则}，该规则在绝大多数诚实参与者可证明地支持执行时，能够实现交易的高效最终确认。对以太坊和Cosmos的实证分析表明，超过 $99\%$ 的区块中验证者参与度超过了所需的 $5/6$ 阈值，这支持了我们设计的实用性。最后，我们提出了一种恢复机制，即使在拜占庭权益高达 $5/9$ 且理性权益高达 $1/9$ 的情况下，该机制也能在一致性违规后恢复安全性和活性，并保证对可证明的客户端损失进行全额赔偿。