  A Federated Byzantine Agreement System is defined as a pair $(V, Q)$
comprising a set of nodes $V$ and a quorum function $Q: V \mapsto 2^{2^{V}}
\setminus \{\emptyset\}$ specifying for each node a set of subsets of nodes,
called quorum slices. A subset of nodes is a quorum if and only if for each of
its nodes it also contains at least one of its quorum slices. The Disjoint
Quorums Problem answers the question whether a given instance of \acrlong{fbas}
contains two quorums that have no nodes in common. We show that this problem is
$\mathsf{NP-complete}$. We also study the problem of finding a quorum of
minimal size and show it is $\mathsf{NP-hard}$. Further, we consider the
problem of checking whether a given subset of nodes contains a quorum for some
selected node. We show this problem is $\mathsf{P-complete}$ and describe a
method that solves it in linear time with respect to number of nodes and the
total size of all quorum slices. Moreover, we analyze the complexity of some of
these problems using the parametrized point of view.
