Stellar representation of extremal Wigner-negative spin states

We use the Majorana stellar representation to characterize maximally nonclassical spin states with respect to the negative volume of the SU(2)-covariant Stratonovich-Wigner quasiprobability distribution. Comparisons are made to alternative definitions of nonclassicality, including anticoherence, the geometric measure of entanglement, and $P$-representability. Despite varying low-dimensional agreement between these definitions, the maximally Wigner-negative states are generally found to disagree with the others, with their higher order constellations not corresponding to a Platonic solid when available, or any other similar geometric embedding. We further find for spin systems with $j \leq \frac{7}{2}$ that random constellations/states are in general not particularly Wigner-negative relative to the maximum. Time permitting, we will also review our proof that all spin coherent states of arbitrary dimension are not positive-definite.