On the Chen Conjecture regarding the complexity of QCSPs

Let A be an idempotent algebra on a finite domain. We combine results of Chen [6] and Zhuk [10] to argue that if Inv(A) satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture (see [7]).