  We introduce twisted quantum $K$-rings, defined via twisted $K$-theoretic
Gromov-Witten invariants. We develop a toolkit for computing relations by
adapting some results about ordinary quantum K rings to our setting, and
discuss some applications, including Ruan-Zhang's quantum $K$-theory with level
structure, and complete intersections inside projective space, confirming some
predictions coming from physics.
  In addition, we formulate a ring-theoretic abelian/non-abelian correspondence
conjecture, relating the quantum K-ring of a GIT quotient $X//G$ to a certain
twist of the quantum K-ring of $X//T$, the quotient by the maximal torus. We
prove this conjecture for the case of Grassmanians, and use this to give
another proof of the Whitney relations of Mihalcea-Gu-Sharpe-Zhou.
