Could the quadratus lumborum cause burning/prickling pain in the upper and lower legs?

Sciatica. You are describing sciatic nerve pain or sciatica, due to compression of the nerve somewhere as it runs from the spinal cord down the leg. Spasm or contraction of the piriformis muscle deep in the buttocks, can cause sciatica. Piriformis syndrome may be causing your symptoms. See your doctor for specific exercises that can help.