As we create more sophisticated quantum systems (including, one day, quantum computers) it becomes imperative to characterize just how much “quantumness” is present in them. However, the exponentiality of Hilbert space, the very feature of Nature that we are trying to exploit in these systems, also poses a significant barrier to verifying quantum behavior. Bell tests offer a powerful solution to this challenge. By performing simple statistical tests on measurement outcomes of spatially separated systems, we can certify not only the presence of quantum behavior, in certain cases we can even characterize the quantum state of the systems, as well as the measurement operators. In recent years, Bell tests have found widespread usage in quantum information processing, from randomness testing protocols to delegated quantum computation.

In this talk, I will survey the recent theoretical progress in using Bell tests to certify high dimensional entanglement --- a setting where the dimensionality is an asymptotically growing parameter. I will also present a new result on testing high dimensional entanglement in the presence of noise (joint work with Rotem Arnon-Friedman). The study of these tests combine a diverse set of ideas and techniques from computer science, cryptography, and physics.