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Lightning Network in Bitcoin ecosystem can bring instant payments and better scalability.

I've read that daughter-chain is one possible solution to bring instant payments into monero, but also that lightning-network would be also possible to implement in monero, once multisig transactions are implemented.

What are differences between lightning network and daughter-chains in regard to instant payments? What pros and cons do they have?

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It's probably better to understand first that daughter-chains are quite different to Lightning Network. A daughter-chain is a side-chain pegged to the main-chain and what's been discussed is having the implementation in the core wallets. This would make it very useful from a user perspective, in my opinion, alongside the more generic use-cases daughter-chains can offer beyond just faster/micro transactions.

...but also that lightning-network would be also possible to implement in monero, once multisig transactions are implemented.

This is incorrect. Multisig (which is now in the code since the v0.12 release) is definitely not all that would be needed to implement Lightning on Monero. There would need to be protocol changes to do things like OP_CheckSequenceVerify.

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