(This is a simple question, with likely a rather involved answer.)
What are the primary obstacles to solve the many-body problem in quantum mechanics?
Specifically, if we have a Hamiltonian for a number of interdependent particles, why is solving for the time-independent wavefunction so hard? Is the problem essentially just mathematical, or are there physical issues too? The many-body problem of Newtonian mechanics (for example gravitational bodies) seems to be very difficult, with no solution for $n > 3$. Is the quantum mechanical case easier or more difficult, or both in some respects?
In relation to this, what sort of approximations/approaches are typically used to solve a system composed of many bodies in arbitrary states? (We do of course have perturbation theory which is sometimes useful, though not in the case of high coupling/interaction. Density functional theory, for example, applies well to solids, but what about arbitrary systems?)
Finally, is it theoretically and/or practically impossible to simulate high-order phenomena such as chemical reactions and biological functions precisely using Schrodinger's quantum mechanics, over even QFT (quantum field theory)?
(Note: this question is largely intended for seeding, though I'm curious about answers beyond what I already know too!)