  We prove that the Schubert structure constants of the quantum $K$-theory ring
of any minuscule flag variety or quadric hypersurface have signs that alternate
with codimension. We also prove that the powers of the deformation parameter
$q$ that occur in the product of two Schubert classes in the quantum cohomology
or quantum $K$-theory ring of a cominuscule flag variety form an integer
interval. Our proofs are based on several new results, including an explicit
description of the most general non-empty intersection of two Schubert
varieties in an arbitrary flag manifold, and a computation of the cohomology
groups of any negative line bundle restricted to a Richardson variety in a
cominuscule flag variety. We also give a type-uniform proof of the
quantum-to-classical theorem, which asserts that the (3-point, genus 0)
Gromov-Witten invariants of any cominuscule flag variety are classical
triple-intersection numbers on an associated flag variety. Finally, we prove
several new results about the geometry and combinatorics related to this
theorem.
