  Complexity theory typically focuses on the difficulty of solving
computational problems using classical inputs and outputs, even with a quantum
computer. In the quantum world, it is natural to apply a different notion of
complexity, namely the complexity of synthesizing quantum states. We
investigate a state-synthesizing counterpart of the class NP, referred to as
stateQMA, which is concerned with preparing certain quantum states through a
polynomial-time quantum verifier with the aid of a single quantum message from
an all-powerful but untrusted prover. This is a subclass of the class stateQIP
recently introduced by Rosenthal and Yuen (ITCS 2022), which permits
polynomially many interactions between the prover and the verifier. Our main
result consists of error reduction of this class and its variants with an
exponentially small gap or a bounded space, as well as how this class relates
to other fundamental state synthesizing classes, i.e., states generated by
uniform polynomial-time quantum circuits (stateBQP) and space-uniform
polynomial-space quantum circuits (statePSPACE). Furthermore, we establish that
the family of UQMA witnesses, considered as one of the most natural candidates,
is in stateQMA. Additionally, we demonstrate that stateQCMA achieves perfect
completeness.
