  For G a complex reductive group and X a smooth projective or convex
quasi-projective polarized G-variety we construct a formal map in quantum
K-theory from the equivariant quantum K-theory $QK^G(X)$ to the quantum
K-theory of the git quotient $QK(X//G)$ assuming the quotient $X//G$ is a
smooth Deligne-Mumford stack with projective coarse moduli space. As an
example, we give a presentation of the (possibly bulk-shifted) quantum K-theory
of any smooth proper toric Deligne-Mumford stack with projective coarse moduli
space. We also provide awall-crossing formula for the K-theoretic gauged
potential under variation of git quotient, a proof of the invariance of certain
K-theoretic Gromov-Witten invariants under (strong) crepant transformation
assumptions, and a proof of a version of the abelian non-abelian
correspondence.
