  The generation and verification of quantum states are fundamental tasks for
quantum information processing that have recently been investigated by Irani,
Natarajan, Nirkhe, Rao and Yuen [CCC 2022], Rosenthal and Yuen [ITCS 2022],
Metger and Yuen [FOCS 2023] under the term \emph{state synthesis}. This paper
studies this concept from the viewpoint of quantum distributed computing, and
especially distributed quantum Merlin-Arthur (dQMA) protocols. We first
introduce a novel task, on a line, called state generation with distributed
inputs (SGDI). In this task, the goal is to generate the quantum state
$U\ket{\psi}$ at the rightmost node of the line, where $\ket{\psi}$ is a
quantum state given at the leftmost node and $U$ is a unitary matrix whose
description is distributed over the nodes of the line. We give a dQMA protocol
for SGDI and utilize this protocol to construct a dQMA protocol for the Set
Equality problem studied by Naor, Parter and Yogev [SODA 2020], and complement
our protocol by showing classical lower bounds for this problem. Our second
contribution is a dQMA protocol, based on a recent work by Zhu and Hayashi
[Physical Review A, 2019], to create EPR-pairs between adjacent nodes of a
network without quantum communication. As an application of this dQMA protocol,
we prove a general result showing how to convert any dQMA protocol on an
arbitrary network into another dQMA protocol where the verification stage does
not require any quantum communication.
