${\mathrm{\Pi }}_2^P$ vs PSpace Dichotomy for the Quantified Constraint Satisfaction Problem

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 ${\mathrm{\Pi }}_2^P$, 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 ${\mathrm{\Pi }}_2^P$-complete.