  Man\v{c}inska and Roberson~[FOCS'20] showed that two graphs are quantum
isomorphic if and only if they are homomorphism indistinguishable over the
class of planar graphs. Atserias et al.~[JCTB'19] proved that quantum
isomorphism is undecidable in general. The NPA hierarchy gives a sequence of
semidefinite programming relaxations of quantum isomorphism. Recently, Roberson
and Seppelt~[ICALP'23] obtained a homomorphism indistinguishability
characterization of the feasibility of each level of the Lasserre hierarchy of
semidefinite programming relaxations of graph isomorphism. We prove a quantum
analogue of this result by showing that each level of the NPA hierarchy of SDP
relaxations for quantum isomorphism of graphs is equivalent to homomorphism
indistinguishability over an appropriate class of planar graphs. By combining
the convergence of the NPA hierarchy with the fact that the union of these
graph classes is the set of all planar graphs, we are able to give a new proof
of the result of Man\v{c}inska and Roberson~[FOCS'20] that avoids the use of
the theory of quantum groups. This homomorphism indistinguishability
characterization also allows us to give a randomized polynomial-time algorithm
deciding exact feasibility of each fixed level of the NPA hierarchy of SDP
relaxations for quantum isomorphism.
