A couple of question clusters that I've always wondered about involve the quantum treatment of systems with a large number of particles.
1) Can we understand the phases of matter - solid, liquid, gas - from quantum mechanics? If so, how? Can we view the bulk matter as a large collection of atoms, treat the atoms as particles, characterize the state of the particles by their positions, then compute some huge tensor product in a high-dimensioinal configuration space, or is some idea in this chain fatally flawed? The "can we" and "how" questions are the basic ones, the more detailed question may be misguided , if so it'd be good to know why it's misguided.
2) We have a piece of matter with some temperature T, regarding it for now as a classical system. If we view it as a quantum system, does it still have a temperature?
Again - some expansions of the fundamental questions, to provide some context, where the expansion questions might have invalid assumptions built into the question.
Suppose this piece of matter with temperature T is a cat. We put the piece of matter (cat) in a box, which is isolated, a closed system. Does the piece of matter (cat) still have a temperature when we think of it as a quantum system? If we wanted to measure a complete set of commuting observables of the piece of matter (i.e. the cat), could we do it in a way that wouldn't change the temperature of the piece of matter (cat)? If necessary, we can regard the temperature as a classical property which we measure before we put the cat in the box, and after we take it out, though the first question is understanding if this is necessary.
If this is possible in principle, what would these commuting observables that don't change the temperature of the piece of matter (cat) be?
We might expect the classical system (cat) to change it's temperature though classical means when put it in the box, due to exothermic reactions for instance. So actually we could allow the measurement of the observables to change the temperature of the system somewhat. We really just don't want to heat up the system too much while observing it, i.e. we don't want to cook the cat.