Bell inequalities for three systems and arbitrarily many measurement   outcomes
  We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can only be reached using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that our inequalities can be used as multipartite dimension witnesses. 
Basile Grandjean, Yeong-Cherng Liang, Jean-Daniel Bancal, Nicolas   Brunner, and Nicolas Gisin
quant-ph