Anticoherent spin states are introduced as quantum states that exhibit maximally nonclassical behaviors which are important in the study of quantum entanglement. Using Majorana representation of spin states, Crann–Pereira–Kribs studied the relation between anticoherent spin states and spherical designs. They proposed a conjecture that a spin- s state is anticoherent to order t if and only if its Majorana representation is a spherical t -design on S ² . In this paper, we prove that this conjecture is true for s = 2.

Relative t-designs in the n-dimensional hypercube Q_n are equivalent to weighted regular t-wise balanced designs, which generalize combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean t-designs on two concentric spheres, in this paper we discuss tight relative t-designs in Q_n supported on two shells. We show under a mild condition that such a relative t-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative t-designs in Q_n supported on two shells are rare for large t.