Fig. presents the polariton dynamics obtained from the Ehrenfest MQC simulations (dotted lines) in the 4level subspace, compared to the numerically exact results (solid lines). It shows that the population dynamics of MQC agrees reasonably well with the exact results. For the 2-level case, our MQC simulation results are also consistent with the exact quantum simulation shown in Fig. (dashed line), which are not presented here. These results provide great confidence of the derived MQC gradient expression.