  We prove a collection of formulas for products of Schubert classes in the
quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This
includes a $K$-theory version of the Seidel representation, stating that the
quantum product of a Seidel class with an arbitrary Schubert class is equal to
a single Schubert class times a power of the deformation parameter $q$. We also
prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal
Grassmannians and Lagrangian Grassmannians, and give a new proof of the known
Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our
formulas have simple statements in terms of quantum shapes that represent the
natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we
give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type
for Lagrangian Grassmannians, and prove a rationality result for the points in
a Richardson variety in a symplectic Grassmannian that are perpendicular to a
point in projective space.
