  The set of pure spin states with vanishing spin expectation value can be
regarded as the set of the less coherent pure spin states. This set can be
divided into a finite number of nested subsets on the basis of higher order
moments of the spin operators. This subdivision relies on the notion of
anticoherent spin state to order $t$: A spin state is said to be anticoherent
to order $t$ if the moment of order $k$ of the spin components along any
directions are equal for $k= 1, 2,\ldots, t$. Most spin states are neither
coherent nor anticoherent, but can be arbitrary close to one or the other. In
order to quantify the degree of anticoherence of pure spin states, we introduce
the notion of anticoherence measures. By relying on the mapping between
spin-$j$ states and symmetric states of $2j$ spin-$1/2$ (Majorana
representation), we present a systematic way of constructing anticoherence
measures to any order. We briefly discuss their connection with measures of
quantum coherence. Finally, we illustrate our measures on various spin states
and use them to investigate the problem of the existence of anticoherent spin
states with degenerated Majorana points.
