Quantum control is actively developed nowadays due to various existing and prospective applications in quantum technologies. For example, in quantum computing it can be used for high fidelity gate generation. We study Turing computability of quantum control problems in the real experimental situation when the number of available controls is finite. For this situation we show that quantum control problems are in general not algorithmically solvable. There is no algorithm (i.e., Turing machine) which could decide if a given quantum problem has globally optimal solution or not. To prove this statement, we develop a technique based on establishing the equivalence between certain quantum control problems and Diophantine equations, polynomial equations with integer coefficients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to different complexity classes. In particular, an example of the control problem involving a two-mode coherent field is shown to be NP-complete. The talk is based on the joint work with D.I. Bondar at https://arxiv.org/abs/1907.10082