Failure of local realism revealed by extremely-coarse-grained measurements

We show that failure of local realism can be revealed to observers for whom only extremely-coarse-grained measurements are available. In our instances, Bell's inequality is violated even up to the maximum limit while both the local measurements and the initial local states under scrutiny approach the classical limit. Furthermore, we can observe failure of local realism when an inequality enforced by nonlocal realistic theories is satisfied. This suggests that locality alone may be violated while realism cannot be excluded for specific observables and states. Small-scale experimental demonstration of our examples may be possible in the foreseeable future.