The final and undeniable proof that there cannot be a classical physics explanation of quantum mechanics will be when a real quantum computer has been constructed and is able to factorimplement and compute a huge number that current classical computers cannot factorquantum algorithm in a time lessfaster than the age of the universewould be possible for a classical computer. This An example of such an algorithm is Grover's Algorithm:
Grover's algorithm is a quantum algorithm for searching an unsorted database with N entries in $O(N^{1/2})$ time and using $O(log N)$ storage space ... In models of classical computation, searching an unsorted database cannot be done in less than linear time (so merely searching through every item is optimal). Grover's algorithm illustrates that in the quantum model searching can be done faster than this; in fact its time complexity $O(N^{1/2})$ is asymptotically the fastest possible for searching an unsorted database in the quantum model. It provides a quadratic speedup, unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts. However, even quadratic speedup is considerable when N is large.
This demonstration of quantum mechanics will be far more convincing than the other interference and entanglement experiments and no-go theorems that Christoph mentions in his answer:
...there's a fundamental disconnect between quantum and classical theories and various no-go theorems that go along with it (Bell, Kochen-Specker, Greenberger–Horne–Zeilinger are probably the most famous ones).
OnCurrent quantum computers are very primitive and have not been able to do any significant computation and certainly cannot currently beat our very fast classical computers since the other hand,quantum computers have very few quantum bits and have very slow cycle times. However a quantum computer with a very slow cycle time implementing Grover's algorithm can beat a faster classical computer if you have enough quantum bits and give it a large enough $O(N)$ problem.
Currently there are many technological problems with building a quantum computer. If we are unableable to overcome all these known problems and we build a quantum computer or if they dothat we are sure should work and we find that it does not work, then we will know that there is something fundamentally wrong with quantum mechanics andand will have the fun of finding the new theory that explains why quantum computers won't work and still gives all the other well tested results that quantum mechanics predicts.