According to THE EXODUS by Richard Elliott Friedman, genetic testing has reported that Levites are genetically diverse, however "Kohens" all derive from one or a few male ancestors in the past couple of thousand years. I understand that all Jews accept that those male ancestors are Aaron, Zadok, so on.
Considerable efforts to find the location of the original altar have been undertaken over past decades to facilitate the resumption of sacrifices.
Once the 3rd Temple normalizes sacrifices, others will presumably follow suit.
That's all well and good. But according to the genetic testing that has actually been done, many different and diverse Arab people from around the region - people who have no history of being identified with the Jews, or of wanting to be - have all 6 of the so-called "Kohanite Markers".
Everybody deeply involved in Jewish religion understands what is at stake: a priesthood must be Aaronic, but unless God indicates a new priesthood, or who really is a descendant of Aaron, no sacrifices could rightly be renewed. If they were anyway, modern Jews would be doing what that Northern Kingdom did in the books of the Kings: NEEDING a priesthood, they simply constituted one, but it wasn't legitimate.
This acts as an effective and real barrier to re-establishing the Temple (which is why God did it that way, and why Jesus' pronouncement of the doom on the Temple was final, and was intended to be).
Unwilling to accept what Jesus' condemnation of the Temple to destruction and the full implications of the New Covenant, some want to rebuild the Old Temple and renew the rituals. They want it so badly they can taste it - but the problem remains: no priesthood.
And so they grasp at straws. The "straw" provided by modern genetic studies is that some people who call themselves descendant from the priests of old have some genetic markers. That is true. For centuries Jews were an inbreeding people in part because of their ostracism and in part because of clannishness. For centuries also, people have claimed to be related to or descended from kings also. Shakespeare's Henry V jokes that everybody he meets claims to be a cousin of his, and if he can't prove it he'll "Fetch it from Japet" - meaning that he'll claim cousinage to the king through their common ancestor Japheth.
The commonality of claims of royal and noble descent, and the willingness of people in earlier ages to attach a "de" or a "von" to their names is - or was - well known in every culture. The highest station in the Jewish imagination is that of "Cohen", and it is not at all surprising that many people in a wandering race ended up with the name "Cohen", for precisely the same reason that a huge percentage of the "de's" in the French diaspora and the "von's" in the German were not, in fact, descended from nobility at all.
Two examples: Honore de Balzac - such a noble name! He put the "de" in his own name, to sound noble and give himself greater "chic". WE know this because he became a famous writer and has been studied. His readers and general public at the time did NOT know that, and simply took him for the nobleman his name announced him to be.
Our American military commander, the "Baron von Steuben", the drill instructor from Germany who trained the Continentals to be able to stand and go toe-to-toe with the British Regulars in line, was neither a Baron nor a "von". But who could fact-check him in 1777?
In precisely the same vein, there are many "Cohens" throughout the Jewish diaspora, but most of them are not REALLY descended from Aaron.
It is true - at the dawning of the age of genetic archaeology, many people with the name "Cohen" were found with a certain genetic pattern, the "Kohanite Six" chromosomal markers. "Look! PROOF!" was proclaimed, and the very hopes were kindled among some. We saw it up thread, where somebody chastised me for being ignorant because the genetic specialness of the Kohein was "proven".
Like I said: grasping at straws, any port in a storm. What has gone unmentioned is that the genetic archaeologists did not stop with the self-identified Kohanim. Yep, that genetic pattern is there. But with further and greater, really ever-expanding, gene pool studies, more and more classified, this "Kohanite Six" marker was found to not be so very special and rare after all. Yes, many self-identified Cohens have it. But many don't. And various clusters of Arabs in the Middle East, including the Saudi peninsula - people who have no memory, tradition or likelihood of every having been associated with the Jewish priesthood, ALSO have the marker.
So if you use that "Kohanite Marker" to re-establish your priesthood, you THINK that you've got the scientific basis for "finding the line of Aaron", but you're as likely to get a Saudi relative of Mohammed as you are to get an actual descendant of Aaron. This pattern is not unique to Jews, is dispersed across the Middle East.
People who are hellbent on getting what they want can dig in on this as they do on everything else, and they will. And they'll even call people who know better "ignorant" because, essentially, the reality of the situation breaks their little toy.
Truth is: if the Aaronic priesthood were ever re-established, it would require a "fire and angels" divine revelation. The genetic studies will get you Egyptians, Palestinians, Saudis, Syrians, Iraqis, Yemenites and Omanis who have no relationship to Aaron other than that which can be "fetched from Shem".
Truth is: Jesus's pronouncing the judgment of doom on the Temple in his generation, and the New Covenant, and all of that, closed a chapter and was intended to, by God.
Truth is: however much men desire to do it their way and to reconstitute what God intentionally broke 2000 years ago, it cannot be done, not truthfully.
Truth is: the "Kohanite" marker isn't really a reliable way to detect the "descendants of Aaron".
Truth is, you can't get there from here. But you WANT TO so very much that you'll pretend you can, and call me ignorant and even a liar.
Christ broke the Temple for a REASON. Truth is, if you grasp at straws and reconstruct an "Aaronic" priesthood using "Kohanite Markers" originally found in self-identified "Cohens", you're going to have to do so ignoring all of those Saudi Arabs, Syrians, Palestinians, Kuwaitis, et al with the same markers who certainly are not descendants of Aaron. This is genetic fool's gold. It's not real. No matter how much you WANT it to be real, it isn't.
God destroyed the Temple for a reason - to make way for the New Covenant, with individuals.
To reconstitute it without an Aaronic priesthood is to take a guess, based on fool's gold, and essentially to recreate the sin of the Northern Kingdom, which wanted its OWN altar so badly it made one.
There are plenty of deeply orthodox Jews who will tell you the same thing (though perhaps without the discussion of the unreality of the Kohanite Six marker as a basis of proving descent from Aaron, because deeply orthodox Jews named Cohen are as likely to want to believe themselves Kohein, and special, and not that some obscure early medieval ancestor adopted the name for the same reason that Major Fred Steuben became the "Baron von Steuben" somewhere on the boat on his way across the Atlantic to a New World that had no internet and no genealogical libraries.
But hey, people are going to believe what you they to believe.
God destroyed the Temple utterly, and wiped out the priesthood for a reason.
Jesus said the law could not be changed a dot or a letter for a REASON.
At least part of the reason was to prevent the Jews from legitimately being able to set a temple back up. You can build a building, but the Aaronic priesthood is gone, and since the Law can't be changed, no OTHER priesthood can be set up to substitute for it.
In other words, it's the New Covenant or bust.
To PRETEND that we can identify Aaron's descendants with genetics and use that as a basis to set up a new Temple, is to repeat the sin of the Northern Kingdom - and to be INCREDIBLY TONE DEAF to what Jesus was doing with the parable of the wine bottles, pronouncing the doom of the Temple, establishing a New Covenant, and then resurrecting himself to make the point.
Jesus came back from the dead.
Aaron and his heirs did not, and can't be brought back by our storytelling.
God did this all for a reason. It is better to spend the time to discern WHY God did it this way, than to pretend that genetic studies can resurrect Aaron so that we can build a Temple again and have it not be a counterfeit. We can't.