Recently, Runeson and Richardson used the spin operators (which are equivalent to the generators of the su(N ) Lie algebra up to a constant) and the SU (N ) Lie group to perform the non-adiabatic mapping dynamics of a N -state vibronic Hamiltonian, and developed the spin-Linearized semi-classical (spin-LSC) approach. In particular, the Stratonovich-Weyl (S-W) transform is used to map an operator in the Hilbert space described by the generators (of the su(N ) Lie algebra) to a continuous function on the Lie group/manifold, resulting in a classical-like Hamiltonian. The S-W transform evaluates the expectation values of the spin operators under the generalized spin coherent states. The generalized spin coherent states are further expressed as a linear expansion of the diabatic electronic states, and the real and imaginary parts of the expansion coefficients are further defined as the conjugate position and momentum variables, leading to an equivalence between the generalized spin-based mapping Hamiltonian and the MMST mapping Hamiltonian. Using the connection between these two Hamiltonians, Runeson and Richardson found the particular choice of zero-point energy parameter for the MMST Hamiltonian as well as an expression of the estimators. With a proposed sampling procedure of the initial conditions that constrains a total population equal to one, spin-LSC effectively propagates the EOMs with a total of 2N MMST variables (2N -2 independent mapping variables when considering the population constraint). This is a reasonable choice for constructing an algorithm for approximate quantum dynamics, as it avoids deriving the EOMs in the generalized Euler angles of the spin coherent state, which is highly non-trivial. However, this choice also brings the potential confusions that the MMST mapping Hamiltonian is a necessary and essential ingredient in the SU (N ) mapping formalism. Further, the EOMs of the spin-LSC approach were not rigorously derived, and there is a lack a rigorous derivation of the timecorrelation function as well. In a related direction, the Generalized Discrete Truncated Wigner Approximation (GDTWA) approach is developed using the generators of the su(N ) Lie algebra. The EOMs for GDTWA were argued as the classical limit of the Heisenberg EOMs of the corresponding operators and generators, where the classical variables for those generators (N 2 -1 of them) obey the equations of the N -dimensional spin precession theory by Hioe and Eberly. However, the EOMs are not rigorously derived in GDTWA approach, and their connection with the MMST-type of EOMs used in spin-LSC remains to be discovered.