  The future progress of semi-device independent quantum information science
depends crucially on our ability to bound the strength of the nonlocal
correlations achievable with finite dimensional quantum resources. In this
work, we characterize quantum nonlocality under local dimension constraints via
a complete hierarchy of semidefinite programming relaxations. In the bipartite
case, we find that the first level of the hierarchy returns non-trivial bounds
in all cases considered, allowing to study nonlocality scenarios with four
measurement settings on one side and twelve (12) on the other in a normal
desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type
inequality that can only be violated when each of the three parties has local
dimension greater than two, hence certifying three-dimensional tripartite
entanglement in a device independent way. Finally, we show how the new method
can be trivially modified to detect non-separable measurements in two-qubit
scenarios.
