QCSP monsters and the demise of the Chen Conjecture

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language $Γ$, QCSP$(Γ)$, where $Γ$ is a finite language over $3$ elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language $Γ$ such that QCSP$(Γ)$ is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class $Θ_{2}^{P}$. Meanwhile, we prove the Chen Conjecture for finite conservative languages $Γ$. If the polymorphism clone of $Γ$ has the polynomially generated powers (PGP) property then QCSP$(Γ)$ is in NP. Otherwise, the polymorphism clone of $Γ$ has the exponentially generated powers (EGP) property and QCSP$(Γ)$ is PSpace-complete.

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