The passuk in Ezra (6:4) says that when Koresh (and later, Daryavesh) gave the Yidden permission to build the second Beis Hamikdash, he told them to make the walls of "נִדְבָּכִין דִּי־אֶבֶן גְּלָל תְּלָתָא וְנִדְבָּךְ דִּי־אָע חֲדַת-three rows of marble stone and one row one wood". The Gemara (Rosh Hashanah 4a) says that the reason he said this was so that if the Yidden rebelled, he can easily burn the Beis Hamikdash, as it has wood in it which makes it flammable. From this we see that Daryavesh, even though he was good in the beginning, soured out. The Gemara asks, that Shlomo Hamelech did the same thing, as the passuk says (melachim I 6:36 in regard to the Azarah, and 7:12 in regard to the Ulam). The Gemara answers that there were three differences between how Shlomo Hamelech built it and how Daryavesh commanded to build it. First of all, Shlomo put his row of wood on top of the three rows of stone, so that even if there was a fire, the bottom of the wall would still be standing, while Daryavesh said to put the wood on the bottom, under the three rows of stone, so that if you burn it, the whole wall collapses. Secondly, Shlomo sunk the wood into the wall, with the stones protruding more than the wood, while Daryavesh made it even. Thirdly, Shlomo covered the wood with plaster, to prevent it from catching on fire, while Daryavesh didn't.
The Passuk in Ezra (6:3) says that when Koresh gave the Yidden permission to build the second Beis Hamikdash, he told them to make it 60 amos wide and 60 amos tall. The question arises, however, that the Mishnah (Middos 4:6) writes that the Heichal was 100 amos wide, long, and tall. The Rambam (Hilchos Beis Habechirah 4:3) writes that this talking about the Heichal built by the Bnei Hagolah, the Yidden who returned from the exile to Bavel and built the second Beis Hamikdash. (This is as opposed to the First Beis Hamikdash, or Hurdus's renovated second Beis Hamikdash, which were both 120 amos tall, as we will soon discuss.)
The simplest answer to this question would seem to be that Koresh did not know exactly how big it was supposed to be, so he said sixty, but he really did not care, and let them make it one hundred. However, the Medrash says (Esther Rabba 1:5) that Hashem was angry at Koresh for dividing the Beis Hamikdash in half, as during the first Beis Hamikdash it was 120 amos tall, and Koresh said to make it sixty. From this it sounds like Koresh says sixty specifically; and did not let more. One can say that when Daryavesh became king and gave the Yidden permission to continue building the Beis Hamikdash (after the construction was halted by Koresh and Achashverosh because of libels by the antisemites), he let them make it one hundred amos tall, however there is another Medrash (Bamidbar Rabba 14:18) which says that the second Beis Hamikdash was actually built sixty by sixty amos. [see Eitz Yosef on the Medrash there, who raises this contradiction between this and the Mishnah in Middos, but does not answer it.]
I have added a new video of the Lishkos on the northern side of the first Beis Hamikdash..
My name is Mendel Lewis.
Hashem said to Yechezkel, "Its reading in the Torah is as great as its building. Go and say it to them, and they will occupy themselves to read the form of it in the Torah. And in reward for its reading, that they occupy themselves to read about it, I count it for them as if they were occupied with the building of it. (Tanchuma tzav 14)