[en] The discrepancy between maximally entangled states and maximally non-classical quantum correlations is well-known but still not well understood. We aim to investigate the relation between quantum correlations and entanglement in a family of Bell inequalities with N-settings and d outcomes. Using analytical as well as numerical techniques, we derive both maximal quantum violations and violations obtained from maximally entangled states. Furthermore, we study the most non-classical quantum states in terms of their entanglement entropy for large values of d and many measurement settings. Interestingly, we find that the entanglement entropy behaves very differently depending on whether N = 2 or $N><!--\; >\; -->2$: when N = 2 the entanglement entropy is a monotone function of d and the most non-classical state is far from maximally entangled, whereas when $N><!--\; >\; -->2$ the entanglement entropy is a non-monotone function of d and converges to that of the maximally entangled state in the limit of large d. (paper)