  The best upper bound for the violation of the Clauser-Horne-Shimony-Holt
(CHSH) inequality was first derived by Tsirelson. For increasing number of $\pm
1$ valued observables on both sites of the correlation experiment, Tsirelson
obtained the Grothendieck's constant ($\mathcal{K}_{G}\approx 1.73\pm0.06$) as
a limit for the maximal violation. In this paper, we construct a generalization
of the CHSH inequality with four $\pm 1$ valued observables on both sites of a
correlation experiment and show that the quantum violation approaching 1.58.
Moreover, we estimate the maximal quantum violation of a correlation experiment
for large and equal number of $\pm 1$ valued observables on both sites. In this
case, the maximal quantum violation converges to $\sqrt{3}\approx1.73$ for very
large $n$, which coincides with the approximate value of Grothendieck's
constant.
