  We give a Chevalley formula for an arbitrary weight for the torus-equivariant
$K$-group of semi-infinite flag manifolds, which is expressed in terms of the
quantum alcove model. As an application, we prove the Chevalley formula for an
anti-dominant fundamental weight for the (small) torus-equivariant quantum
$K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been a
longstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. In
type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials
indeed represent (opposite) Schubert classes in the (non-equivariant) quantum
$K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about the
coefficients in the respective Chevalley formula.
