  In this work we show that bipartite quantum states with local Hilbert space
dimension n can violate a Bell inequality by a factor of order $\sqrt{n}$ (up
to a logarithmic factor) when observables with n possible outcomes are used. A
central tool in the analysis is a close relation between this problem and
operator space theory and, in particular, the very recent noncommutative $L_p$
embedding theory. As a consequence of this result, we obtain better Hilbert
space dimension witnesses and quantum violations of Bell inequalities with
better resistance to noise.
