You don't need Quranic texts for this, you only need to be jewish.
Of course. But it is not a jewish army which enters Jerusalem in 637. On the other hand it is not the Temple which has been rebuilt. It is a prayer building. One have already disputing this point. Sources point that there is a conflict between Jews and Arabs about who has the right on the temple Mount. If it is true, it means that Arabs are not Jewish and that they follow other indication than the jewish one to build something. That they are not jewish is corroborated by the rest of the story : keeping cross on Arab coins, inscriptions with crosses : is that a Jewish custom, crosses and Jesus as prophet in the Dome of the Rock later? I do not think so.
I do not think as well that it was on Jewish instruction that they build something to be stolen by the same Jews of the house they built. And if they did not care, and did it because the Jews have asked them to do so, why expelled the Jews of the Mount then? It is simply because they really wanted to build something and that it did not coming from the Jews. It is probable that the Jews thought to frame them when they saw this building being constructed. Hence the conflict. Unfortunately (for them) they were expelled.
Islamic tradition for example is telling us that it is a Jew allegedly converted to Islam who guided Umar to the area of the Temple Mount ; Sebeos mention an alliance between Jews and Arabs ; we saw in 614, when Persians overtook Jerusalem, that Jews did commit to the war efforts with them. All this need not to be neglected.
Islamic tradition are full of stories to explain events that it ties with the frame Mecca/Medina/Kaba/Muhammad/Zem zem. In the (real) sources there no mention of an "Umar" or anything else in 637 to built on the Mount. Rather, there is a demand of the Arabs where was the Mount and the Jews shown them. I (really) think that the Jews thought that they were rebuilding the Temple for THEM (Jews). And that they could frame the Arabs. One knows the end of the story.
Sebeos mention an alliance between Jews and Arabs ; we saw in 614, when Persians overtook Jerusalem, that Jews did commit to the war efforts with them. All this need not to be neglected.
An alliance where Arabs expelled the Jews from the Mount? I do not call that an "alliance". In 614 the situation is (totally) different; the Persians are in charge. Nor the Jews, nor the Arabs-Persians or anyone else. They all obey to their Persian masters. It happens that they give the city to the Jews to humiliate the Romans (Monophysite, Chalcedonian, Nestorian). But it is not a jewish conquest, it is a Persian one. They REMOVE the city according to the sources from the Jews hand two years later. And what did the Jews? Nothing. They have obeyed.