What properties should asymmetric quorum systems satisfy?

In distributed systems with asymmetric trust, each participant is free to make its own trust assumptions about others, captured by an asymmetric quorum system. This contrasts with ordinary, symmetric quorum systems and threshold models, where trust assumptions are uniformly shared among participants. In the symmetric setting, quorum systems must satisfy the consistency and availability properties to solve key problems like reliable broadcast and consensus. But what properties are needed in the asymmetric setting to solve these problems? We examine this question in both the crash-fault and Byzantine models. In the crash-fault setting, any quorum system satisfying consistency and availability can be transformed into a symmetric one, removing any benefit from asymmetric trust. In the Byzantine model, consistency and availability are not enough to solve reliable broadcast and consensus. Existing approaches overcome this by introducing stronger assumptions. We show that some of these assumptions are overly restrictive, so much so that they effectively eliminate the benefits of asymmetric trust. We introduce a new way to characterize asymmetric problems and, based on this, present protocols for reliable broadcast and consensus that work under weaker assumptions than existing solutions.