I will present a novel perspective on entanglement tests, such as the CHSH inequality and extensions thereof, pioneered by Slofstra and (unknowingly) Gowers-Hatami, that is based on the theory of (approximate) representations of non-abelian groups. We will see how this theory leads to a simple analysis of:

(i) The first family of self-tests (or, two-player entangled games) for arbitrarily high-dimensional entanglement that is robust to a constant fraction of noise. (Joint work with Natarajan, arXiv:1610.03574.)

(ii) The first finite self-test such that for any eps>0, entangled states of dimension poly(1/eps) are both necessary and sufficient to achieve success probability 1-eps. (Joint work with Slofstra, in preparation.)