  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 $\varepsilon$-optimally is at least $\Omega(1/\varepsilon^k)$, for some
$k>0$. Since this function approaches infinity as $\varepsilon$ approaches
zero, this provides a quantitative version of a theorem of the first author.
