If quantum-entangled particles exist (they do) you can use them to coordinate in a way that's impossible within a "hidden variable" model of the universe. You can dig into this by searching for "bell's inequality" or "bell test experiments", there are a few different experiments that have been done many, many times. There's a pretty good Veritasium video that explains one such experiment in non-physicist terms: https://www.youtube.com/watch?v=ZuvK-od647c
The popularized/simplified version is http://en.wikipedia.org/wiki/Free_will_theorem . It's logically impossible for local hidden variables to produce results consistent with this experiment; either the hidden variable must be nonlocal (able to affect two spatially separated experimenters at exactly the same time), or the experimenters' actions must be determined ahead of time (and it's not a hidden variable so much as a hidden constant).
You can definitely disprove negative statements - any positive statement can be expressed as its negation (and vice versa), so to the extent that you can prove positive statements, you can disprove their negations.
>I was unaware it had became possible to disprove a negative.