The unifying idea
Almost every consensus protocol rests on one geometric fact: any two quorums must intersect in at least one node. That shared node carries information from an old decision into a new one, which is what stops the system from contradicting itself.
Why majorities work
A majority quorum is just over half the nodes. Two sets that are each over half cannot be disjoint, so they share at least one member.
- That common node remembers the previously chosen value.
- A new leader contacting any majority is guaranteed to reach it.
- Therefore it cannot accidentally choose a conflicting value.
Generalized quorums
Quorums need not be majorities. A quorum system can use any collection of sets as long as every two of them overlap. Some designs use read quorums and write quorums whose sizes differ, requiring only that read and write sets intersect. For Byzantine settings the overlap must contain at least one honest node, which is why those quorums grow larger.
Key idea
Consensus safety reduces to quorum intersection, every two quorums share a node that carries a past decision forward and prevents contradictions.