  We consider the problem of communicating over a classical-quantum (CQ)
multiple access channel with random classical states non-causally available at
the transmitter, referred to as a QMSTx. QMSTx is a classical-quantum multiple
access analogue of the channel considered by Gelfand and Pinsker in 1980. We
undertake a Shannon-theoretic study and focus on the problem of characterizing
inner bounds to the capacity region of a QMSTx. We propose a new coding scheme
based on \textit{union coset codes} - codes possessing algebraic properties and
derive a new inner bound that subsumes the inner based on IID random coding. We
identify examples for which the derived inner bound is strictly larger.
