The main problem is scaling. In quantum chemistry, you need electron correlation to get accurate results (comparing your ab initio with experiments) and these methods scale quite badly. Somewhat state-of-the-art CCSD(T) (coupled cluster singles and doubles with triple excitations approximated by perturbation theory) scales formally as N^7, with N being the number of electrons in your system (somewhat simplified). Proteins are huge. Even with some of the newest nearly linear-scaling coupled-cluster methods (which utilize GPUs) running calculations on proteins takes extremely long time. And this is just electron correlation methods. We're still using the Born-Oppenheimer approximation, not taking all excitaions into account, using a finite size basis set, ignoring relativistic effects, not thinking about time evolution and (most of the time) not considering (explicit) solvent effects.

We are still a long way from modelling protein folding in solution with 100% quantum mechanics. Luckily, classical molecular dynamics are quite good at modelling biochemical systems (to a degree). There are a ton of errors that a much bigger than quantum (and relativistic) effects anyway.