  This work proposes a $d$-dimensional quantum multi-secret sharing scheme with
a cheat detection mechanism. The dealer creates multiple secrets and
distributes the shares of these secrets using multi-access structures and a
monotone span program. The dealer detects the cheating of each participant
using the Black box's cheat detection mechanism. To detect the participants'
deceit, the dealer distributes secret shares' shadows derived from a randomly
invertible matrix $X$ to the participants, stored in the black box. The Black
box identifies the participant's deceitful behavior during the secret recovery
phase. Only honest participants authenticated by the Black box acquire their
secret shares to recover the multiple secrets. After the Black box cheating
verification, the participants reconstruct the secrets by utilizing the unitary
operations and quantum Fourier transform. The proposed protocol is reliable in
preventing attacks from eavesdroppers and participants. The scheme's efficiency
is demonstrated in different noise environments: dit-flip noise, $d$-phase-flip
noise, and amplitude-damping noise, indicating its robustness in practical
scenarios. The proposed protocol provides greater versatility, security, and
practicality.
