Many areas of quantum physics are complex in different ways. However, I would say things are most complex when you have many particles interacting with each other, so that you cannot focus on what one particle is doing. This happens for instance when you have fluid flows with turbulence or phase changes, or in fact in the early universe, which I study in my research. In fact, these situations would be difficult to study even in classical physics, but there you can always solve the equations if you have a powerful enough computer. But a classical computer is simply not able to do the same for quantum calculations, and therefore you would need a quantum computer, but they do not exist yet. Therefore in cases one needs to first simplify the problem into a form that can be studied with a classical computer.
As Arttu says, many things in QM are complex. The theory of QCD cannot be solved for very low energy interactions, for example, because the strong force of QCD becomes extremely strong at low energy. This means you have to consider not just the exchange of a single gluon, but the exchange of a very large number of gluons – a single gluon can split into two gluons, for example.
So even though we know the theory of QCD quite well, it is extremely hard to solve it in some circumstances.
man, it’s ALL crazily complex to me, but if i had to pick one, then it would probably be quantum entanglement because it really screws with my head and i can’t do the math. here’s a sentence about it that might explain why it makes my head hurt:
“However, there is a profound dispute about whether the objects that are being described by quantum mechanics, in any case under discussion here, already had their real values (e.g., clockwise spin or counterclockwise spin) preset in some way at the instant they became separated, or whether the objects being described by the mathematically indeterminate equations were themselves as indeterminate as were their quantum mechanical descriptions.” huh?