It’s kind of intriguing that predicting the future state of any quantum system becomes almost trivial—assuming you can diagonalize the Hamiltonian. But good luck with that in general. (In other words, a “simple” reference frame always exists via unitary conjugation, but finding it is very difficult.)

Indeed.

It's disconcerting at times, the scope of finite and infinite dimensional linear algebra, especially when done on a convenient basis.