I'm not sure I understood the passage quoted by Alex, but reading it left me with a question. Are there any macroscopic processes which quantum mechanics predicts better than relativistic mechanics? I remember that the standard textbook example of a large-scale phenomenon which classical Newtonian mechanics could not explain, but Einstein's theory of relativity could, was the orbit of Mercury. Is there a similar example for quantum mechanics? Something where our experimental setup is not dealing with individual photons or quarks, but where our macroscopic effect is different under quantum and classical interpretations?