I work as a theorist with my current research interests in "quantum contextuality". For those perusing the question, this is essentially a generalization of non-locality where we can show a quantum system does not admit a hidden-variable model. So in some sense, it's impossible to look at the problem classically.
For further background, contextuality can often be formulated as a resource for quantum computation-- which is the reason why we're interested in it. Our talking points to people are that contextuality is the right way of characterizing what's essentially quantum about a system (many would debate us on that, but that's perhaps for another thread).
What I've found so far is a proposed experiment for testing contextuality: Cabello-2016-Experiment and a description of the implications of a successful contextuality experiment: Winter-2014-Implications.
Most experimentalists tend to focus on entanglement (understandably; it's a lot easier to measure and more concrete to think about) - but are there any experimental groups currently working on contextuality experiments?