Ignoring the issue of political sovereignty, let us focus solely on which religion has the greater claim to that sacred space.
According to the Quran, at one time all of mankind was Muslim; and all men had apostasized into error.
Surah 42:13 ... He enjoined upon Noah and that which We have revealed to you, [O Muhammad], and what We enjoined upon Abraham and Moses and JesusHowever, mankind corrupted the message of Islam; and so Allah raised up one last messenger. Allah sent Jibreel (Gabriel) to a man named Mohammed to restore the original truth to the Arabs, which they were to bring to the world through jihad.
Quran 66:9 Prophet! Make war on the unbelievers ...
According to Islam, since mankind was originally Muslim, all places of true worship must have originally been a mosque.
This is the reason that Muslim Arabs claim justification in taking the sacred areas of whatever lands they seize, and restoring the buildings to what they perceive as their original service to Allah. The "Temple Area" in Jerusalem became the Al-Aqsa (farthest) Mosque, in the city, now renamed Al Quds. Constantinople's Christian Hagia Sofia was defaced and dedicated to Islam. The Hindu Temple of Rama in Ayohya, India destroyed and rebuilt as the Babri Mosque. [The Hindus finally seized it back in 1992] One can only begin to imagine what may soon happen to Notre Dame in Paris.
Against the myths of Islam, Jewish claims are based on history, first person accounts, archeology, and the Bible. The Jews have an embarrassment of riches from which to make their claim to the Mount.