Superior gluteal artery Piriformis is the key to the posterior thigh, and there are many important nerves and arteries that exit the greater sciatic foramen and enter the posterior thigh either above or below piriformis. The superior gluteal artery and nerve are 2 structures that travel through the greater sciatic foramen superior to piriformis. There are many structures that travel through the greater sciatic foramen inferior to piriformis and lie in the posterior thigh. From medial to lateral, these structures are: pudendal nerve, nerve to obturator internus, nerve to quadratus femoris, inferior gluteal artery, posterior femoral cutaneous nerve, inferior gluteal nerve, and sciatic nerve. See Netter plates 502 and 503 for a picture. You should know what nerves and vessels are superior and inferior to piriformis!
None of the other arteries mentioned travel superior to piriformis to reach the posterior thigh. The iliolumbar artery is a branch of the posterior division of the internal iliac. It travels on the posterior body wall to supply blood to the iliacus, psoas major, and quadratus lumborum. The internal pudendal artery is a branch from the anterior division of the internal iliac artery. It leaves the pelvis with the inferior gluteal artery, traveling inferior to piriformis. It enters the perineum through the lesser sciatic foramen to supply muscles, skin, and the erectile bodies there. Finally, the lateral sacral arteries are branches of the posterior division of the internal iliac artery--they are on the posterior body wall and they supply the sacrum, sacral nerve rootlets, meninges, and the nearby muscles.