Why is there no Hallel on Rosh Hashanah and Yom Kippur?
Hallel is the song of festivals, so its absence on the Days of Awe stands out. The Gemara in Arakhin notices. It first explains why some other days have no Hallel. Shabbat is not called a moed, a set festival. Rosh Chodesh is called a moed, but it is not made holy by a ban on work, and a verse in Yeshayahu is read to mean that song belongs to a day made holy that way.
Then it asks about Rosh Hashanah and Yom Kippur. They are called a moed, and they are made holy by the ban on work. By those rules they should have Hallel. The answer is from Rabbi Abahu. The ministering angels asked God why Israel does not sing on these days. God said: is it possible that a King sits on the throne of judgment, with the books of life and the books of death open before Him, and Israel sing songs before Me?
The Rambam says the same in his own words. These are days of repentance, awe and fear, not days of extra joy.
Tosafot (Gemara commentaries from France and Germany, 1100s and 1200s) notice the wording. The angels did not ask why "we" do not sing. They asked why Israel does not. Tosafot take this to mean that the angels themselves do sing on those days.
Purim has no Hallel either. In the Gemara, Rav Nachman says that reading the Megillah is its Hallel, and the Rambam rules that way. So the blessing you just read belongs to the days of joy.
Arakhin 10b,Rambam, Laws of Megillah and Chanukah 3:6,Tosafot on Arakhin 10b,Megillah 14aHallel retells the past. Why would its blessing look to the future?
Hallel is Tehillim 113 to 118, and at the Seder it is about leaving Egypt. But the Gemara finds five things in it (the blessing before Hallel lists them). Three look back: the Exodus, the splitting of the sea and the giving of the Torah. Two look ahead: the revival of the dead, and the hard times before the Messiah ("Not for us, Hashem, not for us").
Abarbanel (Don Yitzchak Abarbanel, a statesman and Torah commentator who was expelled from Spain in 1492) wrote a commentary on the Haggadah in 1496. He gives two readings of Yehallelucha. In the second, it points to that future. This is his reading. The Gemara does not say it.
His clues are in the words. He reads the four verbs yodu, yevarchu, yeshabechu and yefaaru as four groups who praise God: the spiritual beings, the heavens, Israel and the nations. The closing, "King praised with songs," fits a time when God is King over all the earth and is praised by every person.
For the nations he points to the prophets. Yeshayahu says that at the end of days all the nations will stream to God's mountain. Tzefaniah says God will give the peoples a clear speech, so that all of them call in Hashem's name and serve Him shoulder to shoulder.
He adds a point about time. This praise, he says, will not stop all the days of the earth. That is the meaning of the line "from forever to forever You are God." So it is not only about one night.
The words of the blessing do not mention the future. Abarbanel's reading grows out of what Hallel itself says about the redemption to come, and out of those verses.
Pesachim 118a,Zevach Pesach (Abarbanel), on the Haggadah, Hallel,Yeshayahu 2:2-3,Tzefaniah 3:9