On the level complete screen there is a bronze stopwatch or clock stamp/badge alongside the green gems and the yarn ball), but there is no time information as to what triggers it. Is the stamp based off of completion time or some other factor, like deaths? If based on completion time, which co-op player's exit stops the timer? How does it differ between levels and difficulties?
If you look at the checkpoints in some level's there is a time listed on each one of them. This is the time (or sooner) that you should hit these checkpoints. If you hit each checkpoint before the time shown(and make it to the end in a reasonable time) You should get the Stopwatch Stamp.
I am trying to figure out if it is only the time trial levels that this shows up on or if it is all of them. I will try to edit this answer when I find out.