Quantum tunneling does not break the speed of light. It just appears to make something move through a medium faster than classical physics would allow, but it doesn't break the laws of quantum mechanics.
Without more info on that experiment, I'd say there's nothing Star Trek happening. And IIRC, in Star Trek the speed of light isn't violated because "warp speed" is actually bending space-time in order to travel faster than light w/o actually moving faster than light.
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