Even though eCMM and spin-LSC are seemingly derived from very different perspectives, where eCMM is formulated with the MMST Hamiltonian with a total population constraint and spin-LSC is developed based on the SU (N ) mapping formalism, the kernel of eCMM appproach to evaluate the expectation value of the electronic operator turns out to be the Cartesian coordinates ({q n , p n }) expression of the S-W kernel expressed in Eq. 52. In addition, the EOMs used in Spin-LSC and eCMM are identical as well. Considering all of the above, eCMM is an identical approach compared to Spin-LSC. (6). The Symmetric Quasi-Classical (SQC) method uses γ = ( √ 3 -1)/2 for the square window , γ = 1/3 for the triangle window , or a stateand trajectory-specific γ to enforce initial electronic population to be 1 or 0. The initial conditions for the action variable η n (Eq. 122a) is uniformly sampled within a window function, and the angle variable φ n (Eq. 122b) is randomly sampled. The EOMs are identical to Eq. 118 and Eq. 119 with the above mentioned specific choice of γ. Note that the SQC approach assumes the same γ parameter for both window estimators of Â and B, as well as in the EOM.