The state of a system of several quantum particles is an element in the tensor product of Hilbert spaces. In certain situations, the inner product requirement can be relaxed. Quantum states then turn into tensors, and tools and intuition from quantum information, algebraic geometry and algebraic complexity can connect. I will explain the two main traits of application, each leading to an optimization problem. The first is to quantum entanglement - the strong correlations among quantum particles - leading to an optimization over moment polytopes. The second is to many-body physics, where tensor networks are used as an ansatz class to describe the intricate correlations in quantum materials.