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In the same vein, here is something really loopy I remember from uni days.
Note - very simplified explanation glossing over plenty of details.
When you're running a particle accelerator, you slam two particles into each other, and then see two 'jets' of various particles shoot out in opposite directions. Now you can explain the angle of the jets based on the initial particles types, energies etc and QM.
Now the way the jets get started is that after the collision, one initial particle goes one way and the other another. Of course you don't know which particle went which way, and is in which jet (because each jet is a complete mess with lots and lots of particles in it), and this affects the answer QM gives you. And that's the result you get in a real experiment.
But...
You could decide to use an initial pair of particles which were different in some specific way - let's say one is positively charged and the other negatively. Now if you could get a good enough snapshot of the jets, you could simply cancel off particles in either jet, until you found one jet was net positive and the other net negative.
At this point you would know which way the initial particles went. And QM would now give you a different answer for the jet angle. And then what? Which answer do you get? The answer for where you don't know, or the one where you do?
And in either case you have some serious problems.
If you get the 'don't know' answer, then our current understanding of QM would seem to be wrong, which is unfortunate because it works so well.
If you get the 'do know' answer, then the implication is that QM knows in advance that you are going to do this trick, which is flat out nuts - even nutser than QM. And what if you were going to do it and then didn't get around to it?
No one has done this, and (as per my comment above) no one would suggest it would ever be possible in practice, but I find the possibility quite unsettling.
Of course one answer is that the QM (and the laws of physics generally) turn out to forbid you from doing this, so the problem simply doesn't arise.
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