Rotating Platform That Starts and Stops Incrementally
You don't say anything about mass or inertial mass.  

I'm thinking of a stepper motor for this. This will allow you to make precise rotations, like the 90°, though they won't be instantaneous. Depending on the motor's characteristics and the inertial mass you may drive the platform directly or via a belt. Even direct drive shouldn't be a problem for getting the 90° rotation: most steppers will have a multiple of 4 steps per revolution, a 48 steps motor for instance will need 12 steps for the 90°.