  Inspired by computational complexity results for the quantified constraint
satisfaction problem, we study the clones of idempotent polymorphisms of
certain digraph classes. Our first results are two algebraic dichotomy, even
"gap", theorems. Building on and extending [Martin CP'11], we prove that
partially reflexive paths bequeath a set of idempotent polymorphisms whose
associated clone algebra has: either the polynomially generated powers property
(PGP); or the exponentially generated powers property (EGP). Similarly, we
build on [DaMM ICALP'14] to prove that semicomplete digraphs have the same
property.
  These gap theorems are further motivated by new evidence that PGP could be
the algebraic explanation that a QCSP is in NP even for unbounded alternation.
Along the way we also effect a study of a concrete form of PGP known as
collapsibility, tying together the algebraic and structural threads from [Chen
Sicomp'08], and show that collapsibility is equivalent to its
$\Pi_2$-restriction. We also give a decision procedure for $k$-collapsibility
from a singleton source of a finite structure (a form of collapsibility which
covers all known examples of PGP for finite structures).
  Finally, we present a new QCSP trichotomy result, for partially reflexive
paths with constants. Without constants it is known these QCSPs are either in
NL or Pspace-complete [Martin CP'11], but we prove that with constants they
attain the three complexities NL, NP-complete and Pspace-complete.
