Temple rites and the gift of the Holy Ghost are administered by the higher priesthood.
A lot has been written on this subject. In this post, I'm going to try to stick to what the Book of Mormon text says on the subject. For reasons I outline in this post, I believe we can very clearly conclude that Alma the Younger had the Melchizedek priesthood, and by extension I believe it is safe to conclude that the other record-keeping prophets also held the higher priesthood.
Alma 13 is our starting point. The majority of the chapter is about the Melchizedek priesthood. For example, we read:
10 Now, as I said concerning the holy order, or this high priesthood, there were many who were ordained and became high priests of God; and it was on account of their exceeding faith and repentance, and their righteousness before God, they choosing to repent and work righteousness rather than to perish;
11 Therefore they were called after this holy order, and were sanctified, and their garments were washed white through the blood of the Lamb.
12 Now they, after being sanctified by the Holy Ghost, having their garments made white, being pure and spotless before God, could not look upon sin save it were with abhorrence; and there were many, exceedingly great many, who were made pure and entered into the rest of the Lord their God.
This certainly sounds a lot like endowed Melchizedek priesthood holders, doesn't it? Much of this sounds similar to passages in the Doctrine and Covenants which we find in revelations to Joseph Smith years after the Book of Mormon text. I will explore those similarities in future posts.
For now, let's stay focused on the phrase "the holy order," which is one of Alma-2's descriptor's of the higher priesthood and/or temple rites.
In these verses, Alma is talking about other men, not himself. However, he makes it very clear in other sermons that he has been called to this holy order. Here are a few examples:
Alma 5:
44 For I am called to speak after this manner, according to the holy order of God, which is in Christ Jesus;
Alma 6:
8 And Alma went and began to declare the word of God unto the church which was established in the valley of Gideon, according to the revelation of the truth of the word which had been spoken by his fathers, and according to the spirit of prophecy which was in him, according to the testimony of Jesus Christ, the Son of God, who should come to redeem his people from their sins, and the holy order by which he was called.
Alma 8:
4 And he began to teach the people in the land of Melek according to the holy order of God, by which he had been called; and he began to teach the people throughout all the land of Melek.
Other prophets and leaders were ordained to this "holy order" too. See Jacob's words in 2 Nephi 6:2, or Alma 49:30, or Ether 12:10.