Entanglement in non-local games and the hyperlinear profile of groups

We relate the amount of entanglement required to play linear-system non-local games near-optimally to the hyperlinear profile of finitely-presented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play $\epsilon$-optimally is at least $\mathrm{\Omega }(1\mathrm{/}{\epsilon }^k)$, for some $k>0$. Since this function approaches infinity as $\epsilon$ approaches zero, this provides a quantitative version of a theorem of the first author.