(5) It addresses the potential confusions that "the MMST mapping Hamiltonian is a necessary and essential ingredient in the generalized spin mapping formalism". Our theoretical framework clearly shows that one can propagate dynamics with alternative choice of mapping variables, and it is not necessary to introduce the MMST formalism, even though numerically, it might be the most convenient one to implement. The fundamental difference between the SU (N ) mapping and the MMST mapping formalism is also extensively discussed in Appendix F.