These stones under the Western Wall are of impeccable craftsmanship and required seemingly superhuman handling:
My thoughts run to:
Do we know?
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The Western Wall in its entirety was built by King Herod. And yes the stones are quite impressive. This is one of the wonders of the ancient world (I think Josephus calls it that). The top half of the wall was destroyed with the Temple and rebuilt cheaply over the years.
For a full treatment of who built what of the Temple Mount walls, see Ritimyer's book The Quest.
My guide on the Kosel Tunnel Tours (in 2011) explained that they are the work of the simple Jews (not Neviim or leaders) who had a great love and desire to beautify the House of Hashem... He said they were each chiseled by hand and transported through some sophisticated system. I am not sure how much of a source this YU graduate is, but to the best of my recollection these were the words he used.