Measurement in QM is not an all-or-nothing process (and hence there is no such thing as "wave function collapse"). You can measure "just a little bit" by carefully controlling the entanglements of the system you are "measuring". These are called "weak measurements". When you make weak measurements, the probabilities of various outcomes can be non-intuitive because of how the quantum math shakes out, similar to the Bell inequality, except that the entanglements in this case are across time instead of across space. It's cool experimental work, but there's nothing fundamentally new here. It was already well known that the quantum wave function is timeless. This experiment just confirms it.
Bohr would likely have made a big deal over this, because there is no actual measurement on any single system that corresponds to the retrodicted inference, and that matters a great deal in Bohr's quantum ontology.
I'm not a quantum physicist, but I don't get this. Maybe it's just explained in a silly way, but this seems to imply correlation, not "causation" (i.e. the future influencing the present). Correlation isn't that surprising, obviously - if I flip a coin, a friend looks at it, then I look at it, I can predict what my friend saw (in the past) with 100% accuracy.
Obviously I'm missing something.