Qualification occurs every three days (the timer resets when you move to a new league), and checks your progress in the league. To pass it, the progress bar must be filled to the safe threshold, which is indicated by a shield icon:

As a result of qualification, the progress bar is emptied, and you either remain in the current league (if the progress bar is filled to the threshold) or drop to the previous league (if the progress bar is not filled sufficiently).