Looking at this question, I have this related question: there is no doubt that QM gives a very faithful picture of the behaviour of individual particles and atoms.
The question is, is there any evidence that the theory survives intact for systems with a very large number of particles?
If you consider doing a simple textbook quantum experiment that supposedly involves single particles, then a rigorous treatment does need to invoke the validity of quantum mechanics for macroscopic objects as macroscopic measurement devices are going to be involved. E.g. when considering an interferometer, it looks like you are just considering how a single photon moves through the device, bounces off or moves through beam splitters and interferes with itself. But this involves interactions between photons and beam splitters.
When a photon bounces off a beam splitter its momentum changes, the beam splitter absorbs the difference in the momentum. If classical mechanics were exactly valid for macroscopic objects then this momentum change could in principle be extracted from the center of mass momentum of the interferometer (one can imagine a thought experiment where the interferometer is floating in a perfect vacuum in space). This would yield "which way" information which is then inconsistent with observing an interference pattern.
The solution to this paradox is that quantum mechanics also applies to the measurement apparatus, the center of mass momentum can be shown to have a quantum mechanical uncertainty that is much larger than the photon momentum. It's crucial that this uncertainty in the momentum is a real quantum mechanical uncertainty and not merely an uncertainty due to a lack of knowledge of a well defined classical momentum of the macroscopic interferometer.
You could argue organic electronics (which take advantage of orbital hybridization in carbon, a purely QM process) exhibits a macro application of QM (forming the working principle behind the OLED TV).
The carbon molecules used contain a large number particles and exhibit high complexity, requiring a substantial amount of approximation when calculating wavefunctions. This is often achieved through the Hückel model taking advantage of the symmetries in such structures.
Wojciech Zurek has proposed some interesting physics of einselection along these lines. The idea is there is some selection of bases of states for large systems, large number of particles or large action $S = N\hbar$ for $N>>1$. It is not in my opinion a complete idea, for it proposes in some way a superselection type of principle of states based on scale. As yet we do not know what that scaling principle is and how it would really select a basis of states that is stable and "classical."
Physics of the conservation of quantum bits, which I really prefer to see as an invariance principle of entanglement, does tell us that superposition and entanglement phases are conserved, and the decoherent loss of entanglement in a measurement ultimately means this phase is lost to an environment of quantum states. It still exists, but we can't account for it. We have a sense that classical mechanics is wrong within the domain of the very small. It is also somehow built up from quantum systems such as atoms, which means classical mechanics is on some fundamental level false. However, bridging this gap is still a work in progress and has been for many decades.
