the index of the Bloch sphere increases (here, for example, n = 2 and n = 3 in Fig. ), less states are present on the south hemisphere, and the distribution slowly shifts toward the center. This means that the previously used distribution (dashed curves in Fig. ) could potentially have an inconsistent labeling of the angles (the θ 1 distribution belongs to θ N -1 and inversely), as well as an inconsistent distributions for 1 < θ n < N -1 which does not have the right peak on the hemisphere of the Bloch spheres. This demonstrates that the differential phase-space volume element in Eq. 21 is the proper one to consider for the spin coherent state representation of SU (N ), as opposed to the one presented in Ref. 43, which gives incorrect distribution of the θ n variables. We begin with the proposed conjugate relation