  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.
