Per Merriam-Webster, a quorum is, "the smallest number of people who must be present at a meeting in order for decisions to be made."
In the case of a HornetQ cluster it is the smallest number of cluster members which must be able to communicate with each other in order for certain cluster-related decisions to be made. A quorum is a subset of a cluster.
I think currently a quorum is only used when a replicated backup loses the connection to its live server. This situation is described in the last paragraph of section 39.1.2 in the documentation.
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Sorry for the delay. I thought that I had marked it as correct several days ago. My mistake.