At which times did a mainly Jewish sovereign state control Jerusalem?

1) State and sovereignty are political theoretical concepts reflecting historical practices developed in Europe. They entail supremacy of force within a geographic domain. The only state meeting these criteria will be modern.

2) Who is a Jew? Is a fraught cultural and religious question. There is, however, a sovereign state in the Middle East, that functionally controls Jerusalem and has claimed it by an act of that state, with a demographic definition: Israel.

3) Israel claims to have a demographic majority of Jewish people, and effectively enforces its sovereign state control over Jerusalem. It has done so since 1967. Due to the demographic structure of the Ottoman Empire and United Kingdom, and due to the Ottoman Empire's weak state apparatus and limited sovereignty, no other sovereign states are eligible.

4) Questions whose answers fundamentally rely on terms whose definitions are the theoretical products of other social sciences or humanities are weak questions.

I am an Ultra Orthodox Hasidic Jew, very well versed in the Talmud and Jewish tradition, and I am seeing here a lot of confusion, so I will try to clarify.

(Note that I tried as much as possible to reference sources, but not everything has available online sources or even English sources, as many things are just in Hebrew Religious texts, as well as basic Jewish tradition.

For Jews this is simple basics, part of our tradition and culture, just like every American knows the difference between Native Americans and modern Americans, just like this the concept of the Israelites and the Religion is so much part of our culture that it does not make sense otherwise, and the whole discussion is only possible by people from the outside, so in a sense it is hard to bring sources, not so much for the lack of it, but because almost everything is a reference.

Consider the passage in Jeremiah 31 36 where God promises that Israel will be a nation forever, if it's not the Jews then who is it?.)

Israelites vs Canaanites

Regardless of what many historians claim, the Jews and the Israelites never considered themselves as Canaanites, instead they were considered to be the conquers of the Canaanites, see for example Deuteronomy and entire Book of Joshua as well as Kings 1, and according to the Bible the Israelites were commanded by God to kill all the Canaanites, and no jew would consider himself as a Canaanite.

So to answer a question about a Jewish state with Canaanites (as another answer here did) by claiming that Israelites and Canaanites were the same thing makes as much sense as claiming that the Nazis and the Jews were the same thing...

(In fact, Mohammad Abbas (the head of the Palestine Authority) recently declared that the Palestinians are descendants of Canaan, see Times of Israel, thereby attempting to broaden the Israeli-Palestine conflict to the ancient Israelite-Canaanite conflict).

(Although the Patriarchs lived in Canaan in the Canaanite era, and the Israelites also considered themselves as Hebrews which by some interpretations means people that lived in Canaan (as for example in Genesis), they were not part of the Canaanite population, (moreover the Jews believe that Canaan stole the land from Shem the grandfather of Abraham, see in Rashi).

Similarly, there were Canaanites and other ethnicities that remained later in the Israelite period, as can be seen in Judges and in Kings 1 they were not regarded is Israelites, similar to the Native Americans and the Americans that are not considered the same, even though Native Americans also live in America, and even if future historians will try to claim that Americans have evolved from Native Americans).

Regardless of whether one accepts the bible as an actual factual history book, one must admit that the bible is the official Israelite History book, and according to it and according to Jewish Tradition the Canaanites were the worst enemies of the Israelites, and the Israelites considered them as the worst people that have to be wiped out.

Once one realizes this attitude of Israelites to the predecessor, then claiming that the Israelites are descendants of the Canaanites purely on the fact that the Canaanites lived in the land before the Israelites is in my opinion total Ignorance.

Israelites vs Jews

TLDR;

Just like the Germans call themselves Dutch people, although the English speaking world knows them as the German people, the same way the Jews consider themselves as Israelites, and while the English speaking world knows them mostly as Jews for the Jews this is inseparable.

[Source: Jewish Tradition]

Longer Explanation;

The origins of the name differences is from the split between the southern and northern kingdom.

As described in the bible, after the split, the Northern Kingdom (that contained the 10 tribes) called themselves Israel (since they were the majority of Israel), while the Southern Kingdom was known as Judah (which was the name of the Tribe of the Kings of the house of David), however they never considered themselves as separate nations or religions till today (as Jewish tradition goes, and as is evident in many sources, for example in this store in 2 Chronicles).

(More example:

• The Jews as part of mourning the exile of Judah and the destruction of the temple they also morn the exile of the northern kingdom and the 10 tribes, (see the Jewish Kinus Prayer of the 9th of Uv, you can find in Hebrew on Wikisource), and believe that when the Messiah will come all the tribes will be united back (as prophetize by Ezekiel).

• Similarly many Jews consider themselves as descendants of the 10 tribes, for example the Ethiopian Jews-Beta Israel (which officially call themselves Israel) consider themselves as the tribe of Don (see Wikipedia) which was one of the 10 tribes of the northern kingdom, similarly the Jews of Bukharan Jews (according to their tradition) consider themselves as descendants the tribe Zevulun. )

After the Exile of the 10 tribes of Israel, only Judah remained (although some people of the 10 tribes returned and merged with the tribes of Judah), and so at this point although the kingdom of Judah now considered itself again as the Kingdom of Israel (see for example 2 Chronicles), all other nations still called them Judah.

This continued into the Second Temple Era, in which while the Jews considered themselves as Israelites (as in all religious texts such as the Mishna and Talmud etc.) and the land as the Land of Israel (as all religious texts call it, as well as all jews for the past 2000 years), all other nations called them the land of Judah (just like German considers itself to be 'Dutch' and German is just the name that outside people use), until the modern state of Israel which called itself again as Israel.

Given that the land was called as Judah, the Jewish people were mostly called Juda (in German) or Yahud (in Arabic) etc. and in English Jew, (see it also in Wikipedia:Jew).

Interestingly enough, in Russian Jews are called Yevrey, which is apparently a modified version of the name Hebrew which is pronounced in Hebrew as Ivri.

The biggest insult that one can do to a jew, is to claim that he is only a Jew and not an Israelite, though a jew will mostly not even become insulted from such a claim considering it to be as nonsense as a flat earth claim or the blood libel.

Religion

There is no difference between the Religion that the Ancient Israelite held vs today's Judaism (although as with any other religion rituals change somehow, but the core remained the same).

For example, Sacrifice was definitely conducted during the second temple, a time in which everyone agrees that they were already considered as the Jewish religion, and the reason why there is no Sacrifice today is just because the Temple no longer exists, still it is an important part of the Jewish religion and all prayers are centered around the Sacrifices and their absence, and the prayer are considered as temporary substitutions of the sacrifices till the Messiah will come, at which time the Sacrifices will be renewed.

[Source: All Jews know this, and all prayers are centered around the Sacrifice, take any translation of the Jewish Prayer, also you can search online for example Being Jewish or Wikisource, in fact around 1/4 of the Talmud deals exclusively with laws of the Sacrifice although the Talmud was made after the destruction of the temple and in a time that there was no longer Sacrifice.]

(Note, that while the Jewish religion allowed Sacrifices anywhere before the time the temple was built, even though there were dwellings which were similar to the Temple, it is no longer allowed once the temple was built, although many still continued during the time of the first temple to sacrifice outside the temple out of habit it was considered a violation not the law.)

[Source: the bible and Talmud, see also Deuteronomy and etzion ]

Another important point is that the Jews do not distinguish between religion and ethnicity, as only ethnic Jews are required to follow the full religion (besides the 7 Noach commands), and on the other hand any convert is considered just like an ethnic Jew.

[Source: the Bible and Talmud and Jewish Tradition]

There is however a difference in new Rabbinic restrictions and costumes, and some Rabbinic commands (such as the Purim and Hanukkah holidays) which were added at the second temple period, and thus were unknown to the ancient Israelites, and even the Ethiopian Jews were not aware of them being of the 10 tribes, still the rabbanic customs are separate from the original, and even treated differently, and of course it can not be considered as a different religion, just like no one will claim that America has a different law system than it was orignal just because many laws have been made since then.

Unlike Christianity, Judaism believes that the Religion can never be changed not even with the slightest detail, and no prophet ever can change it, as such all the Rabbinical restrictions are treated differently, although the concept of the Rabbinic restriction is based on the original religion, (just like the constitution of every country allows to pass new laws without changing the constitution).

[Source: The Mishnah and Talmud and later commentators such as Maimonides]

The fact that the Northern Kingdom (and sometimes even the Southern) was polytheistic, does not have a relevance to the question anymore than the fact that the later Hasmonean kings belonged to Tzedokim sect which did not follow the Oral Traditions, or the fact that modern Israeli leaders were not observant and even atheists, as the OP didn't ask about Jerusalem being controlled by a fully observant Jewish government, but the question was rather about an independent Jewish government.

Final Answer

As such the answer to the question of the OP is:

1. The entire time of the First Temple (or at least until the last 3 kings).
2. During most of the period of the Hasmonean.
3. The modern state of Israel.