That's a classical ambiguity, not a quantum ambiguity. It would be like if I placed a camera that recorded when cars arrived but I only gave you information on when it detected a car and at what time and no other information, not even providing you with the footage, and asked you to derive which car came first. You can't because that's not enough information.

The issue here isn't a quantum mechanical one but due to the resolution of your detector. In principle if it was precise enough, because the radiation emanates from different points, you could figure out which one is first because there would be non-overlapping differences. This is just a practical issue due to the low resolution of the measuring device, and not a quantum mechanical ambiguity that couldn't be resolved with a more precise measuring apparatus.

A more quantum mechanical example is something like if you apply the H operator twice in a row and then measure it, and then ask the value of the qubit after the first application. It would be in a superposition of states which describes both possibilities symmetrically so the wavefunction you derive from its forwards-in-time evolution is not enough to tell you anything about its observables at all, and if you try to measure it at the midpoint then you also alter the outcome at the final point, no matter how precise the measuring device is.