I think the "Stalin hated the jews" is a confused mix of outright anti-communist propaganda and a misunderstanding or deliberate misrepresentation of later soviet policy.

Usually "the doctors plot" gets brought up, which was an alleged conspiracy involving nine doctors, six of whom were jewish. After a moderate amount of kerfuffle nothing came of it, no one was prosecuted, and nothing happened.

There are allegations that Stalin was going to put Jewish Soviets in camps. There is absolutely nothing to these arguments. No documents, nothing. Just "well my unlce talked to some guy" bs.

Anti-semitism was a very serious crime during most of the ussr's history.

During the cold war Israel"s "aliyah" bullshit was used by a bunch of ethnic Jewish Soviets as an excuse to defect. There was a lot of back and forth between the Soviet state and Soviet Jews about emigrating, and the US and Israel were happy to use it as a bludgeon. Afaik it had little to do with anti-Semitism and everything to do with emigration to Israel creating a brain-drain among a subset of Soviet citizens, with fully half of the people who emigrated to Israel actually switching flights half-way and defecting to the US, taking their education and technical knowledge with them. Israel actually imposed a bunch of "prove you're Jewish" rules when it turned out a lot of people who got visas to move to Israel had no intention of doing so and hadn't had any real connection to Jewish religion or culture for decades, they just had "Jewish" as their national classification on their id paperwork because their parents or grandparents were Jewish.