The GHZ/Mermin experiment refined the Bell argument into an "all-or-nothing" experiment with three parties. In that scenario, there are no probabilistic tests. Alice, Bob and Charlie each receive one particle (entangled, of course) and make a local decision to measure one of two quantities of that particle: x or y, each with a binary outcome. The decision to measure x or y is local and beyond the light cone of influence of each.
No matter what they chose to measure, the results are strongly correlated. So much so, that it is clear that you cannot predict every outcome of what the three parties measure. There just isn't a single pre-determined logically consistent assignment of outcomes before the actual measurement (there is a parity restriction of the outcomes).
But the measurements are consistent anyway. So, either there is faster-than-light communication which enforces the consistency, or indeed the outcomes cannot be decided until one of them measures something. It just isn't mathematically possible.
This is also independent of the physical theory by the way. You don't need Schroedinger, Dirac, or anything. Any physical theory that predicts measurements just like QM would will have this restriction. So hidden variables are incompatible with locality (no faster-than-light communication).
The experiments are fine and useful for constraining any better interpretation.
The problems are in their interpretation and in the interpretation of much of the formalism of quantum mechanics and quantum field theory.
There are far more possible interpretations that match the known experimental facts than it would appear from university textbooks.
The problem is that for now it is not possible to decide which of them is right, because nobody has been able to work out their complete consequences by creating a complete coherent quantum theory.
There are a lot of alternative interpretations that sound more plausible than the mainstream interpretation, but none of them has been developed enough to be able to prove that they could explain everything.
This is a subject that requires a book-length treatment. There are actually some such books, but they only show the very serious flaws of the mainstream interpretation of quantum theory, without being able to propose an adequate complete replacement. Criticism is always easier than discovering a better solution.
The only certitude is that the current quantum mechanics/quantum field theory are not the final theories of physics and whoever believes that physics is solved is as naive as those who believed the same thing at the end of the 19th century.
The GHZ/Mermin experiment refined the Bell argument into an "all-or-nothing" experiment with three parties. In that scenario, there are no probabilistic tests. Alice, Bob and Charlie each receive one particle (entangled, of course) and make a local decision to measure one of two quantities of that particle: x or y, each with a binary outcome. The decision to measure x or y is local and beyond the light cone of influence of each.
No matter what they chose to measure, the results are strongly correlated. So much so, that it is clear that you cannot predict every outcome of what the three parties measure. There just isn't a single pre-determined logically consistent assignment of outcomes before the actual measurement (there is a parity restriction of the outcomes).
But the measurements are consistent anyway. So, either there is faster-than-light communication which enforces the consistency, or indeed the outcomes cannot be decided until one of them measures something. It just isn't mathematically possible.
This is also independent of the physical theory by the way. You don't need Schroedinger, Dirac, or anything. Any physical theory that predicts measurements just like QM would will have this restriction. So hidden variables are incompatible with locality (no faster-than-light communication).