  We study the computational complexity of quantum discord (a measure of
quantum correlation beyond entanglement), and prove that computing quantum
discord is NP-complete. Therefore, quantum discord is computationally
intractable: the running time of any algorithm for computing quantum discord is
believed to grow exponentially with the dimension of the Hilbert space so that
computing quantum discord in a quantum system of moderate size is not possible
in practice. As by-products, some entanglement measures (namely entanglement
cost, entanglement of formation, relative entropy of entanglement, squashed
entanglement, classical squashed entanglement, conditional entanglement of
mutual information, and broadcast regularization of mutual information) and
constrained Holevo capacity are NP-hard/NP-complete to compute. These
complexity-theoretic results are directly applicable in common randomness
distillation, quantum state merging, entanglement distillation, superdense
coding, and quantum teleportation; they may offer significant insights into
quantum information processing. Moreover, we prove the NP-completeness of two
typical problems: linear optimization over classical states and detecting
classical states in a convex set, providing evidence that working with
classical states is generically computationally intractable.
