Despite having an unnatural definition, $\mathsf{StoqMA}$ plays a central role in Hamiltonian complexity, e.g., in the classification theorem of the complexity of Hamiltonians by Cubitt and Montanaro (SICOMP 2016). Moreover, it lies between the two randomized extensions of $\mathsf{NP}$, $\mathsf{MA}$ and $\mathsf{AM}$. Therefore, understanding the exact power of $\mathsf{StoqMA}$ (and hopefully collapsing it with more natural complexity classes) is of great interest for different reasons. In this work, we take a step further in understanding this complexity class by showing that the Stoquastic Sparse Hamiltonians problem ($\mathsf{StoqSH}$) is in $\mathsf{StoqMA}$. Since Stoquastic Local Hamiltonians are $\mathsf{StoqMA}$-hard, this implies that $\mathsf{StoqSH}$ is $\mathsf{StoqMA}$-complete. We complement this result by showing that the separable version of $\mathsf{StoqSH}$ is $\mathsf{StoqMA}(2)$-complete, where $\mathsf{StoqMA}(2)$ is the version of $\mathsf{StoqMA}$ that receives two unentangled proofs.

翻译：尽管其定义并不自然，但$\mathsf{StoqMA}$在哈密顿量复杂性中扮演着核心角色，例如Cubitt和Montanaro (SICOMP 2016)关于哈密顿量复杂性的分类定理。此外，它位于$\mathsf{NP}$的两个随机化扩展$\mathsf{MA}$和$\mathsf{AM}$之间。因此，理解$\mathsf{StoqMA}$的确切能力（并希望将其归结为更自然的复杂性类）对不同方面都具有重要意义。在这项工作中，我们通过证明逗留稀疏哈密顿量问题（$\mathsf{StoqSH}$）属于$\mathsf{StoqMA}$，进一步加深了对这一复杂性类的理解。由于逗留局域哈密顿量问题是$\mathsf{StoqMA}$-难的，这意味着$\mathsf{StoqSH}$是$\mathsf{StoqMA}$-完全的。我们通过证明$\mathsf{StoqSH}$的可分离版本是$\mathsf{StoqMA}(2)$-完全的这一结果来补充上述结论，其中$\mathsf{StoqMA}(2)$是接收两个非纠缠证明的$\mathsf{StoqMA}$版本。