This common question is not easily answered. Many muscles contribute to a vertical leap, including your entire lower body, core, and some of your upper body musculature. For example, although the calves are commonly given credit for leaping ability, if you only had your calf muscles and not your thigh and hip muscles, you probably wouldn't be able to jump very high. Also, your upper body is important in helping to generate momentum for the vertical jump, making it vitally important as well. In summary, the most important muscle for for improving your vertical is all of them! However, when choosing only one muscle to focus on, a strong gluteus maximus is crucial to good vertical. This is due to its force production capabilities since it is the largest muscle in the body, and if this muscle is weak, it can disrupt lower-extremity mechanics which can lead to decreased overall force production and poor jumping performance.
More Answers from National Academy of Sports Medicine