In this work, we show that verifying the order of a finite group given as a black-box is in the complexity class QCMA. This solves an open problem asked by Watrous in 2000 in his seminal paper on quantum proofs and directly implies that the Group Non-Membership problem is also in the class QCMA, which further proves a conjecture proposed by Aaronson and Kuperberg in 2006. Our techniques also give improved quantum upper bounds on the complexity of many other group-theoretical problems, such as group isomorphism in black-box groups.

翻译：本文证明，对于以黑盒形式给出的有限群，验证其阶数属于复杂性类QCMA。该结果解决了Watrous在其2000年关于量子证明的奠基性论文中提出的公开问题，并直接推导出群非成员判定问题同样属于QCMA类，从而进一步证实了Aaronson与Kuperberg于2006年提出的猜想。我们的技术方法还为许多其他群论问题（如黑盒群中的群同构判定）提供了改进的量子计算复杂度上界。