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Nakamoto’s longest-chain consensus paradigm now powers the bulk of the world’s cryptocurrencies and distributed finance infrastructure. An emblematic property of longest-chain consensus is that it provides probabilistic settlement guarantees that strengthen over time. This makes the exact relationship between settlement error and settlement latency a critical aspect of the protocol that both users and system designers must understand to make informed decisions. A recent line of work has finally provided a satisfactory rigorous accounting of this relationship for proof-of-work longest-chain protocols, but those techniques do not appear to carry over to the proof-of-stake setting. This article develops a new analytic approach for establishing such settlement guarantees that yields explicit, rigorous settlement bounds for proof-of-stake longest-chain protocols, placing them on equal footing with their proof-of-work counterparts. Our techniques apply with some adaptations to the proof-of-work setting where they provide improvements to the state-of-the-art settlement bounds for proof-of-work protocols. 

Nakamoto proof-of-work ledger consensus currently underlies the majority of deployed cryptocurrencies and smart-contract blockchains. While a long and fruitful line of work has succeeded to identify its exact security region---that is, the set of parametrizations under which it possesses asymptotic security---the existing theory does not provide concrete settlement time guarantees that are tight enough to inform practice. In this work we provide a new approach for obtaining concrete and practical settlement time guarantees suitable for reasoning about deployed systems. We give an efficient method for computing explicit upper bounds on settlement time as a function of primary system parameters: honest and adversarial computational power and a bound on network delays. We implement this computational method and provide a comprehensive sample of concrete bounds for several settings of interest. We also analyze a well-known attack strategy to provide lower bounds on the settlement times. For Bitcoin, for example, our upper and lower bounds are within 90 seconds of each other for 1-hour settlement assuming 10 second network delays and a 10% adversary. In comparison, the best prior result has a gap of 2 hours in the upper and lower bounds with the same parameters.