Title: Third Temple
Content:  Initially, the Emperor Hadrian granted permission to rebuild the temple but then changed his mind. The forces of Simon bar Kokhba captured Jerusalem from the Romans in 132 CE, and construction of a new temple continued. The failure of this revolt led to the writing of the Mishna, as the religious leaders believed that the next attempt to rebuild the temple might be centuries away and memory of the practices and ceremonies would otherwise be lost. As punishment for the revolt, the Romans renamed the city to Aelia Capitolina and the province to Syria Palaestina and Jews were prohibited in the city except for the day of Tisha B'av. However, the Rabbis that survived persecution (see Ten Martyrs) were allowed to continue their school in Javnia, as long as they paid the Fiscus Judaicus.
Title: Demographic history of Jerusalem
Content:  prominent citizens deported to Babylon (Biblical sources only) ; 539 BCE: Cyrus the Great conquers Babylon, allowing Babylonian Jews to return from the Babylonian captivity to Jerusalem and rebuild the Temple (Biblical sources only, see Cyrus (Bible) and the Return to Zion) ; 530 BCE: The Second Jewish Temple was rebuilt, on the same Temple Mount as the first Jewish Temple. ; 350 BCE: Jerusalem revolts against Artaxerxes III, who retakes the city and burns it down in the process. Jews who supported the revolt are sent to Hyrcania on the Caspian Sea. ; 332–200 BCE: Jerusalem capitulates to Alexander 
Title: History of Jerusalem
Content:  According to the Bible, and perhaps corroborated by the Cyrus Cylinder, after several decades of captivity in Babylon and the Achaemenid conquest of Babylonia, Cyrus II of Persia allowed the Jews to return to Judah and rebuild the Temple. The books of Ezra–Nehemiah record that the construction of the Second Temple was finished in the sixth year of Darius the Great (516 BCE), following which Artaxerxes I sent Ezra and then Nehemiah to rebuild the city's walls and to govern the Yehud province within the Eber-Nari satrapy. These events represent the final chapter in the historical narrative of the Hebrew Bible. During this period, Aramaic-inscribed "Yehud coinage" were produced – these are believed to have been minted in or near Jerusalem, although none of the coins bear a mint mark.
Title: Second Temple Judaism
Content:  The period of the First Temple ended in 586 BCE when the Babylonian king Nebuchadnezzar captured Jerusalem, destroyed the Temple of Solomon, and deported the elite of the population to Babylon (the "Babylonian exile"). In 539 BCE Babylon itself fell to the Persian conqueror Cyrus, and in 538 BCE the exiles were permitted to return to Yehud medinata, as the Persian province of Judah was known. The Temple is commonly said to have been rebuilt in the period 520–515 BCE, but it seems probable that this is an artificial date chosen so that 70 years could be said to have 
Title: Cyrus the Great in the Bible
Content:   Cyrus the Great was the founder of the Achaemenid Empire and king of Persia from 559-530 BC. He is venerated in the Hebrew Bible for conquering Babylon and liberating the Jews from captivity. He is mentioned 23 times by name and alluded to several times more. According to the Bible, Cyrus the Great, king of the Achaemenid Empire, was the monarch who ended the Babylonian captivity. In the first year of his reign he was prompted by God to decree that the Temple in Jerusalem should be rebuilt and that such Jews as cared to might return to their land for this purpose. Moreover, he showed his interest in the project by sending back with them the sacred vessels which had been taken from the First Temple and a considerable sum of money with which to buy building materials. The existence of the decree has been challenged.
Title: Ezra 6
Content:  The Persian court searched the royal archive to investigate the historical claim of the Jews for rebuilding the temple, first in Babylon, according to Tattenai's suggestion (Ezra 5:17) but they found a scroll containing Cyrus's edict in Ecbatana (modern Hamadan in northern Iran, former capital of Median Empire.) Darius the king of Persia issued a decree supporting the temple building project.
Title: Pharisees
Content:  of Jacob" (Deut. 33: 4). After the destruction of the First Temple, Jews believed that God would forgive them and enable them to rebuild the Temple – an event that actually occurred within three generations. After the destruction of the Second Temple, Jews wondered whether this would happen again. When the Emperor Hadrian threatened to rebuild Jerusalem as a pagan city dedicated to Jupiter, in 132, Aelia Capitolina, some of the leading sages of the Sanhedrin supported a rebellion led by Simon Bar Kosiba (later known as Bar Kokhba), who established a short-lived independent state that was conquered by the Romans in 135. With 
Title: Temple in Jerusalem
Content:  this event to this day as the central theme of the non-biblical festival of Hanukkah. The temple was rededicated under Judah Maccabee in 164 BCE. During the Roman era, Pompey entered (and thereby desecrated) the Holy of Holies in 63 BCE, but left the Temple intact. In 54 BCE, Crassus looted the Temple treasury. Around 20 BCE, the building was renovated and expanded by Herod the Great, and became known as Herod's Temple. It was destroyed by the Romans in 70 CE during the Siege of Jerusalem. During the Bar Kokhba revolt against the Romans in 132–135 CE, Simon bar Kokhba and Rabbi Akiva 
Title: Cyrus Cylinder
Content:  The Bible records that some Jews (who were exiled by the Babylonians), returned to their homeland from Babylon, where they had been settled by Nebuchadnezzar, to rebuild the temple following an edict from Cyrus. The Book of Ezra (1–4:5) provides a narrative account of the rebuilding project. Scholars have linked one particular passage from the Cylinder to the Old Testament account: "From [?] to Aššur and [from] Susa, Agade, Ešnunna, Zabban, Me-Turnu, Der, as far as the region of Gutium, the sacred centers on the other side of the Tigris, whose sanctuaries had been abandoned for a long time, I returned the images of the 
Title: Solomon's Temple
Content:  carrying the blood of a sacrificial lamb and burning incense. According to the Bible, the Temple not only served as a religious building, but also as a place of assembly for the Israelites. The Jews who had been deported in the aftermath of the Babylonian conquest were eventually allowed to return and rebuild their temple — known as the Second Temple. But the building no longer housed the Ark, as it had disappeared. There is a general agreement that a ritual structure existed on the Temple Mount by the point of the Babylonian siege of Jerusalem, however serious doubts remain in attributing it 

Question: who rebuilt the temple after the babylonian captivity?
Answer: Herod.