What is the device taken from Alex Trusks at the end of Hypercube?

It may be much more than a simple recording device. As shown, time moves differently inside, yet all of Jerry's watches are perfectly synchronized. We know they were taken from different versions of Jerry at different times, so if they were functioning as normal watches, you'd have wildly different times on each watch, as well as much more time having passed.

The fact that they all line up to the "6:06:59" reading with Kate suggests that Alex Trusk's necklace was some kind of stabilizing device. The watches (and time) stabilized and ran normally when in proximity to the device. This is also supported by the fact that at the end, when the room is disintegrating, the only one that remains is Kate, who took the device from Alex's neck after her death.

This would explain why Simon ran into multiple versions of Jerry and the others and experienced more time, but Kate did not.

Of course, that doesn't explain how what seems like hours passed for Kate after she met up with Alex (and was in proximity to the device), yet the watches still only show 6 minutes and 59 seconds have actually passed.

I suppose it's possible that the device had SOME stabilizing effect, but not completely. And as the cube imploded, normal time from the outside universe began leaking in.

Or, it could just be that Jerry's watch only counts time for the outside universe. Modern watches count time using the oscillation frequency of quartz when exposed to an electric charge, so if somehow that were unaffected by the weird time frame of the cube, it would explain it. In effect, the watches would only tick when exposed to time from the outside universe. If normal time seeped through to the cube at predictable intervals (and they were increasing), you could probably use that to calculate when the cube would implode. Hence, the scientist making that "6:06:59" calculation.

In fact, it could be both of those explanations together. If so, and Alex Trusk were hiding in the tesseract, it would make sense that they would send in an operative to retrieve it. She developed a way to stabilize (and ultimately control) the tesseract, and based on her hiding, I doubt she sent the government instructions. They'd need to retrieve it to figure out how it worked.

Or it could be just weird, timey-wimey stuff. In the end, WHO KNOWS. :-P