We study subspaces of the vector space of spin-s states which have the property that all of their elements are scalar multiples of anticoherent subspaces. We give a characterization of these anticoherent subspaces in terms of certain joint higher rank numerical ranges. We also look at examples of subspaces of the vector spaces of spin-s states all of whose elements have Majorana representations that are spherical designs. Concrete examples of such subspaces are constructed using group theory. Some open problems are presented.

Coauthor: Connor Paul-Paddock