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Currently, checking for double spends is very computationally expensive, so consensus algorithms, like POW, were put in place to more easily prevent double spends among other things.

Ignoring the fact that a consensus algorithms allows for other essential things to be done, like making sure that new cryptocurrency can only be mined at a fixed rate; if there was a way to very efficiently detect double spends in an incoming blockchain and the protocol was changed so it will reject a blockchain that has double spends and it will accept accept a blockchain with no detected double spends, then will there be no need for consensus algorithms?

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Yes, if you assume that the central challenge of a distributed system in an antagonistic environment doesn't exist, hypothetically creating such a distributed system is indeed much easier!

Currently, checking for double spends is very computationally expensive[…]

Detecting doublespends is trivial. You just need to recognize that two transactions are in conflict because they spend the same funds.
The actual challenge of the doublespending problem is to get a network of equal peers to converge on one canonical transaction order. In fact, Bitcoin's most notable achievement is to propose a practical solution to the doublespending problem.

the protocol was changed so it will reject a blockchain that has double spends and it will accept accept a blockchain with no detected double spends, then will there be no need for consensus algorithms?

So, sure, if you assume that there unexplainedly is already consensus on how to solve the principal problem of operating a decentralized currency system, you no longer need to figure out how to achieve consensus. However, that's not a particularly useful starting point.

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  • What do you mean when you say, "not a particularly useful starting point"?
    – CCS
    May 29, 2021 at 23:51
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    The entire reason for the existence of PoW is because it solves the consensus problem - as consensus is needed to prevent double spends. If your assumption is that you already don't need that, and thus already have a consensus system, there is indeed not much interesting to build anymore. May 31, 2021 at 3:15
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PoW is merely a "thing to agree upon". You can choose any arbitrary thing or system to agree upon. Time, for example. Television programming line up. Sports scores. Bus schedule. Humidity fluctuations. Stock market changes. Or, anonymous electronic transactions. If you remove the money/greed aspect then it becomes waaaay easier to mitigate double transactions ("spends").

Think of a chatroom for example. That meets all the previously mentioned descriptors. Anonymous, electronic transactions of low monetary value. Now then, everyone participating in the chat room records an identical copy of all the transactions, right? So imagine if each time you reply to a specific user, you increment a unique sequential index. So to the recipient it looks like this: From Satoshi: @CCS hello fren! (Index #42)

Then when you reply to them the index is incremented, so they see from you: From CCS: @Satoshi long time no see, fren! (#43)

Since this little chain of incremental identifiers consists of only two parties, they will never run into many of the problems in an adversarial network. Each participant knows what next index is expected and if a message appears that deviated from that, everybody will know there is a problem.

Here we see everyone transacting independently without a third party and they are able to verify the order of transactions. If the transactions additionally included a sum (chksum), then each transaction would also be able to verify immutability. (error correction)

If one of my messages to you contains a PGP encrypted message containing my credit card number and a purchase order, then I have sent you an electronic funds transfer on a public distributed verifiable share ledger that involves no wasted PoW or mining nonsense. []

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