In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states and, time allowing, describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our recent preprint, we provide high-probability upper bounds in different regimes on the injective norm of real and complex Gaussian random tensors, which corresponds to lower bounds on the geometric entanglement of random quantum states, and to bounds on the ground-state energy of a particular multispecies spherical spin glass model. Our result is a first step towards solving an important folklore question in quantum information.