League qualifications
If you go to
Help > Learn to Play > Player Ratings
in the app, it explains that it is based on a moving average of your last 50 scores, with some exceptions. The score ranges are subject to change according to the help file, but are, at the time of this writing:
Not rated - <100
Bronze - 100 - 165
Silver - 165 - 240
Gold - 240 - 325
Platinum - 325 - 425
Diamond - 425+