  Trigonometric cubature rules of degree $t$ are sets of points on the torus
over which sums reproduce integrals of degree $t$ monomials over the full
torus. They can be thought of as $t$-designs on the torus. Motivated by the
projective structure of quantum mechanics, we develop the notion of $t$-designs
on the projective torus, which, surprisingly, have a much more restricted
structure than their counterparts on full tori. We provide various
constructions of these projective toric designs and prove some bounds on their
size and characterizations of their structure. We draw connections between
projective toric designs and a diverse set of mathematical objects, including
difference and Sidon sets from the field of additive combinatorics, symmetric,
informationally complete positive operator valued measures (SIC-POVMs) and
complete sets of mutually unbiased bases (MUBs) (which are conjectured to
relate to finite projective geometry) from quantum information theory, and
crystal ball sequences of certain root lattices. Using these connections, we
prove bounds on the maximal size of dense $B_t \bmod m$ sets. We also use
projective toric designs to construct families of quantum state designs.
Finally, we discuss many open questions about the properties of these
projective toric designs and how they relate to other questions in number
theory, geometry, and quantum information.
