We stopped using musical instruments in the synagogue after the Temple was destroyed, as a sign of mourning. We used them before that time (see Psalm 150). When the Temple is rebuilt, does halacha say we can go back to using them? For that matter, do we know if they went back to using them after the Second Temple was built?
Very brief background: In ancient days, musical instruments were used in worship: [Mishna, Arachin 2:3 - Sefaria: https://www.sefaria.org/Mishnah_Arakhin.2.3?lang=bi&with=all&lang2=en]: There are never less than two harps, nor more than six. There are never less than two flutes, nor more than twelve... And they did not play on a bronze pipe but on a reed pipe... Nor was anything but a single pipe used for the finale, because it makes a pleasant finale.
Then the Temple was destroyed, and the Talmud seemed to forbid all music, anytime: [Gittin 7a. (Sefaria: https://www.sefaria.org/Gittin.7a?lang=bi)] They sent the following question to Mar Ukva: From where do we derive that song is forbidden in the present, following the destruction of the Temple? He scored parchment and wrote to them: “Rejoice not, O Israel, to exultation, like the peoples” (Hosea 9:1). The Gemara asks: And let him send them a response from here: “They do not drink wine with a song; strong drink is bitter to them who drink it” (Isaiah 24:9), indicating that song is no longer allowed. The Gemara answers: If he had answered by citing that verse, I would say that this matter applies only to instrumental music, in accordance with the previous verse: “The mirth of tabrets ceases, the noise of them who rejoice ends, the joy of the harp ceases” (Isaiah 24:8); however, vocal song is permitted. Therefore, Mar Ukva teaches us that all types of song are forbidden. This was later codified [Rambam, Hilchot Ta’aniyot 5:14; Shulchan Aruch in OH 560:3]
This was softened to allow song. Sotah 48a: "Rav Huna nullified all types of song... Later, when Rav Ḥisda came and belittled this prohibition, people began to sing again."
But we can't mourn on Shabbat/Yom Tov! So the prohibition against musical instruments on those days rests on the prohibition against "completing" an object on Shabbat [Eruvin 104a]. Playing an instrument "completes" it [Rambam, Hil’chot Shabbat 23:4; Shulchan Aruch, Orach Chayyim 338:1]. Also, one might end up fixing (or tuning?) the musical instrument. [Mishna, Beitza 36b]