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Blockchain protocols differ in fundamental ways, including the mechanics of selecting users to produce blocks (e.g., proof-of-work vs. proof-of-stake) and the method to establish consensus (e.g., longest chain rules vs. Byzantine fault-tolerant (BFT) inspired protocols). These fundamental differences have hindered "apples-to-apples" comparisons between different categories of blockchain protocols and, in turn, the development of theory to formally discuss their relative merits. This paper presents a parsimonious abstraction sufficient for capturing and comparing properties of many well-known permissionless blockchain protocols, simultaneously capturing essential properties of both proof-of-work (PoW) and proof-of-stake (PoS) protocols, and of both longest-chain-type and BFT-type protocols. Our framework blackboxes the precise mechanics of the user selection process, allowing us to isolate the properties of the selection process that are significant for protocol design. We demonstrate the utility of our general framework with several concrete results: 1. We prove a CAP-type impossibility theorem asserting that liveness with an unknown level of participation rules out security in a partially synchronous setting. 2. Delving deeper into the partially synchronous setting, we prove that a necessary and sufficient condition for security is the production of "certificates," meaning stand-alone proofs of block confirmation. 3. Restricting to synchronous settings, we prove that typical protocols with a known level of participation (including longest chain-type PoS protocols) can be adapted to provide certificates, but those with an unknown level of participation cannot. 4. Finally, we use our framework to articulate a modular two-step approach to blockchain security analysis that effectively reduces the permissionless case to the permissioned case.