We all see folks with low back pain and gait abnormalities. It is active during single limb support during stance phase of gait on the contralateral side (along with the external oblique) to elevate the ilium. This is coupled with the ipsilateral anterior fibers of the gluteus medius and minimus pulling the iliac crest toward the stable femur.
We found this FREE FULL TEXT while doing some quadratus lumborum research. It reminds us about things like scleratogenous pain (pain arising from tissues of like embryological origin with a common nerve innervation, like tendon, bone, muscle, etc) and other triggers for low back pain. We have needled this ligament with good result. Remember that this is an individual ligament making up a portion of the middle layer of the thoracolumbar fascia, and is not an aponeurosis of the lumbocostal fibers of the quadratus lumborum.
photo from: https://musculoskeletalkey.com/treatment-of-the-patient-with-chronic-pain/