Thinking about it logically (my 2 cents logic)
The quarter squats would help improve cleans but you would also have to train the other movements of the clean or snatch to such as the pull, hang, explosive hip extension as it is an explosive movement compared to the squats.
The quarter squats compared to the full squats would involve different muscles being activated.
Top half range of motion: From just above thighs parallel to standing, the loading demand will be greater on your hip extensors. This means that your glutes, adductor magnus (inner thigh), and to a lesser extent, hamstrings, are most activated in this range of motion.
Bottom half range of motion: From about thighs parallel and below, the loading demand will be greater on your knee extensors. This means that your quads are most activated in this range of motion.
With jumping or sprinting acceleration, gaining strength in the limited knee bent that takes place and then doing explosive work in that same range would help a lot.
I think I remember they did study that showed partial squats providing better strength to jumping than full squats.