The positioning of the bar will dictate what mechanics must be used to maintain lifting efficiency. The mechanics will dictate what musculature is used. Succinctly, more vertical squat techniques use the quads and glutes as the primary movers while the low bar puts a premium on the posterior chain (particularly the hamstrings) for hip drive. The low bar squat also has a balanced anterior/posterior force at the knee because the focus on the hamstring contraction pulls back on the knee. In contrast, the high bar has more of an anterior stress since the quads are the primary movers and attach at the front of the knee at the patella. There is some quadriceps involvement in the low bar squat, but not nearly as much as the high bar squat. There is a little hamstring involvement in the high bar (see below), but not nearly as much as the low bar squat.
The forward knee placement in the front and high bar squats result in an acute knee angle. An acute knee angle means that the hamstrings — crossing the knee and hip joints — are contracted and slackened. If the hamstrings are fully contracted, then they cannot contract to help extend the hip out of the hole. This means that in vertical squatting styles, there is no hamstring involvement out of the hole, and limited involvement throughout the ascent. However, note the slight torso difference between the front and high bar squats; the inclination in the high bar style provides more of an angle that allows some hamstring tension. The upper two-thirds of the ascent of a high bar squat can have assistance by the hamstrings to hold the back angle in place (the same way that they do with the low bar or a deadlift). This makes sense as the position can re-apply tension on the hamstrings once the knees are no longer acutely flexed. I have felt this when high bar squatting, but you can see it during these hellacious sets that Max Aita did at California Strength.
To clarify the hamstring involvement during the high bar: the descent occurs, the knees flex acutely at the bottom in the hole, the ascent begins with zero hamstring tension due to knee flexion, as the knee angle opens the hamstrings can receive tension, and there is hamstring tension during the ascent. Note that this tension is not a primary mover due to the torso angle.
In contrast, the low bar squat maintains tension throughout the descent, creates a stretch reflex to “bounce” off the tense hamstrings, and then utilizes the hamstrings to extend the hip. During the ascent, the quads obviously extend the knee, but they help create balance around the knee so that the hip drive doesn’t tip the torso forward (the torso obviously needs to maintain it’s angle out of the hole). Since this style of squatting is dependent on the hamstrings, the body’s positioning — particularly the knees — is much more important. If the knees shift forward at the bottom, then hamstring tension will decrease and will result in no bounce whatsoever. Discussing other faults in the low bar squat leaves the scope of this post.