  Let A be an idempotent algebra on a finite domain. We combine results of
Chen, Zhuk and Carvalho et al. to argue that if 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 QCSP(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.
  We examine in closer detail the situation for domains of size three. Over any
finite domain, the only type of PGP that can occur is switchability.
Switchability was introduced by Chen as a generalisation of the already-known
Collapsibility. For three-element domain algebras A that are Switchable, we
prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible.
The significance of this is that, for QCSP on finite structures (over
three-element domain), all QCSP tractability explained by Switchability is
already explained by Collapsibility.
  Finally, we present a three-element domain complexity classification
vignette, using known as well as derived results.
