Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a p

Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a proton can also change over time (usually due to interactions with other particles, such as stray radio waves passing through your experiment). Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected! Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.