Benefit of Schnorr signature

[Oshosondy]

I have been going through what Taproot is for a while now, it is the ungrade which is having approaches from bitcoin communities about how it will be successfully activated this year, probably the activation will be in November if speedy trial is supported and signalled by majority of miners. I do not know much about the activation and how it will be activated, but I want to know the benefit of Taproot and Schnorr signature.

1.  Will Schnorr signature nullifies the threat of quantum computing?
2. How are transactions going to be, will bitcoin users be able to create a multisig transaction with it and with low fee paid?
3. Is Schnorr signature part of elliptic curve cryptography or another aspect entirely?

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[NotATether]

1.  Will Schnorr signature nullifies the threat of quantum computing?

No. Because Schnorr signatures use elliptic curve cryptography (to answer your third question as well) - secp256k1 group for Bitcoin specifically, and they rely on the discrete logarithm problem being hard in that curve, a quantum computer might be able to solve it and thus break Schnorr signatures in polynomial time.

2. How are transactions going to be, will bitcoin users be able to create a multisig transaction with it and with low fee paid?

Some new multisignature schemes (Musig1/2) will use Schnorr signatures but with the additional feature that public keys can be aggregated together and have this aggregate public key put in the transaction which improves privacy. Since only one public key is in the transaction, the size is reduced and therefore less fees are paid, but otherwise nothing else changes with respect to fees.

[Oshosondy]

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Thanks for the fast response, I noticed Taproot will make use of fee almost similar to segwit (bc1) but slightly higher, if multisig transactions can use it, this will create means for people to use segwit with low fee paid for transactions, this is a great thing as it is amazing.

[Pmalek]

I read about Schnorr signatures a few months ago. If I remember correctly, signature aggregation is just one important feature. The privacy improvements will make it impossible for others to know that you are using a multisignature address. Since the signatures are merged into one, it's also not going to be possible to see what kind of multisig is being used, or who signed the transaction.

[pooya87]

Don't forget about "batch verification"[1]. From what I understand you can basically verify all signatures in one block for example (assuming they are all using Schnorr) in one operation instead of individually. And that gives a decent speed up[2] which is very useful in block verification.
You can find more information in BIP-340.

[1] https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki#Batch_Verification
[2] https://github.com/bitcoin/bips/blob/master/bip-0340/speedup-batch.png