The Quantified Constraint Satisfaction Problem is the problem of evaluating a sentence with both quantifiers, over relations from some constraint language, with conjunction as the only connective. We show that for any constraint language on a finite domain the Quantified Constraint Satisfaction Problem is either in $\Pi_{2}^{P}$, or PSpace-complete. Additionally, we build a constraint language on a 6-element domain such that the Quantified Constraint Satisfaction Problem over this language is $\Pi_{2}^{P}$-complete.

翻译：量化约束满足问题是：在来自某个约束语言的关系上，以合取为唯一连接词，对同时包含两种量词（全称和存在量词）的句子进行真值判定的问题。我们证明，对于有限定义域上的任意约束语言，量化约束满足问题要么属于 $\Pi_{2}^{P}$，要么是 PSpace-完全的。此外，我们构造了一个基于6元素定义域的约束语言，使得该语言上的量化约束满足问题是 $\Pi_{2}^{P}$-完全的。