Basically Wheeler's delayed choice plus one, done with measurements separated by large distances, and measurement of which path information done in a completely separate location long after the results should have arrived at the detectors. 12 quantum physicists worked on this paper.
I can do a better job of explaining what is going on and why this experiment is significant, so let's do it. First, in erasure experiments there is the concept of "choice" affecting the outcome of the experiment. That is, whether we choose to obtain which path information or not decides the outcome of the experiment. WHEN whe choose to obtain that information is irrelevant. In most of these experiments, the "choice" is said to be made by the insertion or removal of 50/50 beam splitters. Basically the beam splitters make the choice. So quantum physicists decided they'd toy around with the concept of choice, and when the choice is being made.
The first experiment most here know about is the Delayed Choice Quantum eraser, seen here:
In this picture I have circled the beamsplitters, which are the points at which the choice is made to erase which path information or not. This choice is made after the other photon hits D0, yet the result at D0 correlates with the results at the other detectors.
Next was Wheeler's delayed choice experiment, seen here:
Here the choice made by the beamsplitter is just before the final detectors. With this beamsplitter in place, you get a diffraction pattern. With the beamsplitter gone, you get a clump pattern. This is always true even if you remove/place the beamsplitter (make the choice) after the photon has already entered the experiment.
Finally, they have taken Wheeler's concept a bit further in this particular setup. First, they begin with the standard erasure mechanism:
This is basically the same setup as in Wheeler's delayed choice. Left as depicted here in the picture (final beamsplitter in place), you will always see the diffraction pattern. What this experiment does, however, is say "what if we leave the last beamsplitter in place, but somehow move the choice of how the particle behaves to a completely different location?"
This idea is realized in the following:
They leave the beamsplitter in place, but the choice of whether or not we see a diffraction pattern is made in a completely different location, after the photon has already entered the experiment on the right of "S". That's why they call it a "causally disconnected choice". And, when they chose to collect which path data over to the left, in another building, they got the clump pattern on the right at D1/D2. When they chose not to collect which path data, they got the interference. The which path measurement is done long after the experiment on the right of "S" is done and over with. They just proved that the choice of whether or not to collect which path data is all that matters, and that choice has no physical connection to the result. Both the choice and the measurement are outside the light cone of the result. Thus confirming the predictions of MBT.
And so they came to the following conclusions:
"The fact that it is possible to decide whether a wave or particle feature manifests itself long after—and even space-like separated from—the measurement teaches us that we should not have any naive realistic picture for interpreting quantum phenomena."
"Our results demonstrate that the viewpoint that the system photon behaves either definitely as a wave or definitely as a particle would require faster-than-light communication. Because this would be in strong tension with the special theory of relativity, we believe that such a viewpoint should be given up entirely. "