What was the function of the Jewish priesthood after the Temple was destroyed? Many have opined that the priesthood slowly disappeared as an institution as a result of the lack of any real cultic utility, but priestly lines were maintained for centuries afterward, and there are some indications that they maintained some function, particularly in the synagogues. Greek and Roman rulers set out to dilute and handicap priestly authority long before 70 CE, but as a social class they still maintained a great deal of influence (so much so that Herod had the newly appointed High Priest, Arisobulus III, drowned after the masses expressed joy at his appointment; and Josephus appealed proudly to his priestly lineage). Where they basically a Jewish aristocracy? What was their relationship to the rabbis? Without the temple the law became the new liturgy, as far as the literature reveals. Did they compete with each other over ecclesiastical clout? The opposition of the rabbis to many practices associated with the synagogues may suggest this, but what kind of priestly lineage could be averred in the diaspora?
These are some of the more annoying questions that keep me up at night (when my eight-month-old does not).