Note that the EOMs in Eq. 117 and Eq. 119 are identical to the EOMs commonly used in the MMST mapping formalism. These EOMs are also used in the spin-LSC approach, by the argument 43 that they are Hamilton's EOMs of H (Eq. 118). Here, we rigorously prove that they are equivalent to the EOMs in Eq. 98, thus can be derived as the linearization approximation from the exact quantum Liouvillian (Eq. 85 and Eq. 87). The 2N MMST mapping variables are subject to a constraint given in Eq. 56. Further, there is an overall phase factor among the {q n , p n } variables which does not influence the dynamics (see Eq. 46). Thus, there are still 2N -2 truly independent variables, in agreement with the 2N -2 generalized Euler angle variables {ϕ n , θ n } or the 2N -2 conjugate variables {ϕ n , Θ n }. The EOMs with {q n , p n } in Eq. 117 are indeed analytically simpler than the EOMs with {ϕ n , θ n } in Eq. 113, making them more appealing for practical implementations.