  Quantum nondeterministic distributed computing was recently introduced as
dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall,
Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of quantum
proofs and local communication, nodes on a network verify a global property of
the network. Fraigniaud et al. showed that, when the network size is small,
there exists an exponential separation in proof size between distributed
classical and quantum verification protocols, for the equality problem, where
the verifiers check if all the data owned by a subset of them are identical. In
this paper, we further investigate and characterize the power of the dQMA
protocols for various decision problems.
  First, we give a more efficient dQMA protocol for the equality problem with a
simpler analysis. This is done by adding a symmetrization step on each node and
exploiting properties of the permutation test, which is a generalization of the
SWAP test. We also show a quantum advantage for the equality problem on path
networks still persists even when the network size is large, by considering
``relay points'' between extreme nodes.
  Second, we show that even in a general network, there exist efficient dQMA
protocols for the ranking verification problem, the Hamming distance problem,
and more problems that derive from efficient quantum one-way communication
protocols. Third, in a line network, we construct an efficient dQMA protocol
for a problem that has an efficient two-party QMA communication protocol.
  Finally, we obtain the first lower bounds on the proof and communication cost
of dQMA protocols. To prove a lower bound on the equality problem, we show any
dQMA protocol with an entangled proof between nodes can be simulated with a
dQMA protocol with a separable proof between nodes by using a QMA
communication-complete problem introduced by Raz and Shpilka (CCC 2004).
