Title: An Optimal Multilevel Quorum System for Probabilistic Consensus

Abstract: We present the notion of a multilevel, slashable quorum system, where an application can obtain gradual levels of assurance that a certain value is bound to be decided (or "finalized") in a global consensus procedure, unless a large number of Byzantine processes are exposed to slashing (that is, penalty on staked assets). Our construction is a highly parameterized generalization of quorum systems based on finite projective spaces, with asymptotic high availability and optimal slashing properties. In particular, we show that any quorum system whose ground elements are disjoint subsets of nodes (e.g. "commmittees" in committee-based consensus protocols) has asymptotic high availability under very reasonable conditions, a general proof with significance of its own. Under similarly relaxed conditions, we show that our construction has asymptotically optimal slashing properties with respect to message complexity and process load; this illustrates a fundamental trade off between message complexity, load, and slashing. Our multilevel construction allows nodes to decide how many "levels" of finalization assurance they wish to obtain, noting that this functionality, if applied to a proof-of-stake blockchain, can be seen either as (i) a form of an early, slashing-based, probabilistic block finalization; or (ii) a service for reorg tolerance.