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