We extend the notion of anticoherent spin states to anticoherent subspaces. An anticoherent subspace of order t is a subspace whose unit vectors are all anticoherent states of order at least t. We use Klein's description of algebras of polynomials which are invariant under finite subgroups of SU (2) on C 2 to provide constructions of anticoherent subspaces. We discuss applications of this idea to the entanglement of n qubit symmetric states. Furthermore, we show a connection between the existence of these subspaces and the properties of the higher-rank numerical range for a certain products of spin observables. We also note that these constructions give us subspaces of spin states all of whose unit vectors have Majorana representations which are spherical designs of order at least t.

For the classes of synchronous, binary constraint systems, and XOR nonlocal games, we show that near-optimal finitedimensional quantum strategies with arbitrary states are approximate representations of their affiliated nonlocal game algebra. We also show that finite-dimensional approximate representations of these nonlocal game algebras are close to near-optimal strategies where the players employ a maximally entangled state. As a corollary, we show that near-optimal quantum strategies are close to a near-optimal quantum strategy that uses a maximally entangled state.