  The product of two Schubert classes in the quantum K-theory ring of a
homogeneous space X = G/P is a formal power series with coefficients in the
Grothendieck ring of algebraic vector bundles on X. We show that if X is
cominuscule, then this power series has only finitely many non-zero terms. The
proof is based on a geometric study of boundary Gromov-Witten varieties in the
Kontsevich moduli space, consisting of stable maps to X that take the marked
points to general Schubert varieties and whose domains are reducible curves of
genus zero. We show that all such varieties have rational singularities, and
that boundary Gromov-Witten varieties defined by two Schubert varieties are
either empty or unirational. We also prove a relative Kleiman-Bertini theorem
for rational singularities, which is of independent interest. A key result is
that when X is cominuscule, all boundary Gromov-Witten varieties defined by
three single points in X are rationally connected.
