The blockchain data structure maintained via the longest-chain rule---popularized by Bitcoin---is a powerful algorithmic tool for consensus algorithms. Such algorithms achieve consistency for blocks in the chain as a function of their depth from the end of the chain. While the analysis of Bitcoin guarantees consistency with error $2^{-k}$ for blocks of depth $O(k)$, the state-of-the-art of proof-of-stake (PoS) blockchains suffers from a quadratic dependence on $k$: these protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that depth $Theta(k^2)$ is sufficient. Whether this quadratic gap is an intrinsic limitation of PoS---due to issues such as the nothing-at-stake problem---has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctness. We give an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longest-chain rule and achieve, in broad generality, $Theta(k)$ dependence on depth in order to achieve consistency error $2^{-k}$. In particular, for the first time, we show that PoS protocols can match proof-of-work protocols for linear consistency. We analyze the associated stochastic process, give a recursive relation for the critical functionals of this process, and derive tail bounds in both i.i.d. and martingale settings via associated generating functions.
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