[Cronos]

I would say Christianity is a continuation of 2nd Temple Judaism. It is a sect that AROSE from 2nd temple judaism just as the Pharisees, Sadducees, Samaritans etc. were 2nd temple Judaic sects

With the destruction of the 2nd temple, the sects that relied on the temple, the sacrifices and the priests (Cohens) were destroyed as their very core was destroyed. This was reinforced with the Bar kochkba revolt where Judaic peoples slaughtered the pagan gentiles in Cyrene and Cyprus - thereby bringing the wrath of the Romaoi Empire on themselves and getting themselves banned from their Holy Land

2nd temple Judaism died in 69 AD and it was succeeded by 3 sects - or daughter religions

1. the Jesus movement - or Christianity
2. Pharisees - Modern day Judaism - centered around the synagogue, led by Rabbis not priests and with no blood sacrifices
3. Samaritans who had their own center of worship at Mt Gerizim

The Sadduccee and Essene sects died out

Net-net -> Christianity and Modern-day Judaism are sister religions both arising out of 2nd Temple Judaism. In fact the Jewish canon was closed well into the Christian era at the Council of Jamnia where Christocentric texts were edited out.

[Cronos]

Let me emphasise though - this joint birth does not mean modern Judaism is in any way "false" - just that, neutrally speaking ,it is on par with Christianity and parochially speaking it has the flaw of not recognizing the Messiah (that's me as a Christian speaking of course)

None of this justifies in any way any kind of persecution of one by the other.