"Negative probabilities" is being imprecise -- as you allude to, what we really mean are quantum mechanical amplitudes that are out-of-phase relative to each other, such that we get constructive and destructive interference when you convert them into concrete probabilities. (Feynman also acknowledges this lack of precision in terminology, but ultimately this text was not intended to be rigorous scientific proof but rather building intuition for this problem that he was deeply interested in.)
I believe Feynman's discussion of hidden variables is a reference to the EPR paradox (see: Einstein's infamous quote that "God does not play dice") and the various Bell tests (which at this point in time had experimentally demonstrated that hidden-variable theories were inadequate for describing QM). If you continue in the paper, he then goes on to describe one of those experiments involving entangled photons.
In particular, what we definitely can't do is generate random numbers for measurements of individual particles while assuming that they're independent from each other. So now we have to consider the ensemble of particles, and in particular we need to consider the relative phases between each of them. But now we're getting back to the same exponential blowup that caused us to run into problems when we tried to simulate the evolution of the wavefunction from first principles.
I believe Feynman's discussion of hidden variables is a reference to the EPR paradox (see: Einstein's infamous quote that "God does not play dice") and the various Bell tests (which at this point in time had experimentally demonstrated that hidden-variable theories were inadequate for describing QM). If you continue in the paper, he then goes on to describe one of those experiments involving entangled photons.
(I believe he described this setup: https://en.wikipedia.org/wiki/Bell_test#A_typical_CH74_(sing...)
In particular, what we definitely can't do is generate random numbers for measurements of individual particles while assuming that they're independent from each other. So now we have to consider the ensemble of particles, and in particular we need to consider the relative phases between each of them. But now we're getting back to the same exponential blowup that caused us to run into problems when we tried to simulate the evolution of the wavefunction from first principles.