  The question of how large Bell inequality violations can be, for quantum
distributions, has been the object of much work in the past several years. We
say that a Bell inequality is normalized if its absolute value does not exceed
1 for any classical (i.e. local) distribution. Upper and (almost) tight lower
bounds have been given for the quantum violation of these Bell inequalities in
terms of number of outputs of the distribution, number of inputs, and the
dimension of the shared quantum states. In this work, we revisit normalized
Bell inequalities together with another family: inefficiency-resistant Bell
inequalities. To be inefficiency-resistant, the Bell value must not exceed 1
for any local distribution, including those that can abort. This makes the Bell
inequality resistant to the detection loophole, while a normalized Bell
inequality is resistant to general local noise. Both these families of Bell
inequalities are closely related to communication complexity lower bounds. We
show how to derive large violations from any gap between classical and quantum
communication complexity, provided the lower bound on classical communication
is proven using these lower bound techniques. This leads to
inefficiency-resistant violations that can be exponential in the size of the
inputs. Finally, we study resistance to noise and inefficiency for these Bell
inequalities.
