  Byzantine quorum systems provide higher throughput than proof-of-work and
incur modest energy consumption. Further, their modern incarnations incorporate
personalized and heterogeneous trust. Thus, they are emerging as an appealing
candidate for global financial infrastructure. However, since their quorums are
not uniform across processes anymore, the properties that they should maintain
to support abstractions such as reliable broadcast and consensus are not
well-understood. It has been shown that the two properties quorum intersection
and availability are necessary. In this paper, we prove that they are not
sufficient. We then define the notion of quorum subsumption, and show that the
three conditions together are sufficient: we present reliable broadcast and
consensus protocols, and prove their correctness for quorum systems that
provide the three properties.
