The polynomial hierarchy has been widely studied in classical complexity theory. In this paper, we will generalize some commonly known results about the polynomial hierarchy to a version of the hierarchy extended to promise problems. This paper proposes new definitions of existential and universal operators for classes of promise problems. Applying these to BQP, we recover the hierarchy proposed by Gharibian et al. (MFCS 2018). Moreover, using our definition, we give an easy proof of the collapse of this hierarchy under a Karp-Lipton-like scenario, which was an open question for the original definition of Gharibian et al.

翻译：多项式层级在经典复杂性理论中已被广泛研究。本文将一些关于多项式层级的已知结果推广到承诺问题扩展的层级版本。本文提出了承诺问题类存在算子与全称算子的新定义。将这些算子应用于BQP时，我们恢复了Gharibian等人（MFCS 2018）提出的层级结构。此外，利用我们的定义，我们给出了在类似Karp-Lipton场景下该层级折叠的简单证明，而这一结果在原Gharibian等人的定义中仍是一个开放问题。