Cross-subspace alignment (CSA) codes are used in various private information retrieval (PIR) schemes (e.g., with secure storage) and in secure distributed batch matrix multiplication (SDBMM). Using a recently developed $N$-sum box abstraction of a quantum multiple-access channel (QMAC), we translate CSA schemes over classical multiple-access channels into efficient quantum CSA schemes over a QMAC, achieving maximal superdense coding gain. Because of the $N$-sum box abstraction, the underlying problem of coding to exploit quantum entanglements for CSA schemes, becomes conceptually equivalent to that of designing a channel matrix for a MIMO MAC subject to given structural constraints imposed by the $N$-sum box abstraction, such that the resulting MIMO MAC is able to implement the functionality of a CSA scheme (encoding/decoding) \emph{over-the-air}. Applications include Quantum PIR with secure and MDS-coded storage, as well as Quantum SDBMM.