Quick answer: The best squat variation for developing bigger quads is the upright close stance high bar squat in weightlifting shoes. This version makes the quads perform a lot of work while the weight and the weight that can be lifted is quite heavy.
Why the high bar squat?
The high bar squat allows the lifter to be more upright which places more stress on the quadriceps. If you were to do a classic low bar squat, you will find out that you are going to lift more weight, but you are also going to bend over more which shifts the stress to the posterior chain muscles (hamstrings and glutes).
Why not the front squat? It allows you to be even more upright.
Because the front squat is not very comfortable to do for high reps and does not allow you to lift as much weight as the high bar back squat. Of course, the front squat will develop your quadriceps too, but it’s not as optimal as the back squat. The front squat also places more stress on the knees which can be an issue, if you increase your workload.
Why close stance?
The close stance is the best for quads because it reduces the involvement of the inner tight muscles known as adductors. In other words you are cutting help from some muscle groups in order to overload the quadriceps.
Why weightlifting shoes?
Weightlifting shoes allow you to stay even more upright which further increases the stress on the quadriceps. Those kind of shoes have a small wooden heel and also help you squat deeper. The negative side is that weightlifting shoes stress the knees more.
What about sissy squats?
Sissy squats are a stupid exercise that for the most part does only one thing – stress you knees while failing to develop any kind of appreciable strength. For the most part it’s safer to do other forms of squats in order to avoid injuries to the connective tissues of the knee joints.
Why not hack squats?
The hack squat is a decent quadriceps exercise but similar to the sissy squat it’s notorious for killing knee joints. Most people experience problems from doing hacks squats in the long run.