“Ecash is the fastest way to send money on the internet” - Robin Linus, creator of BitVM and BitStream.

He’s right. With Pay-to-Pubkey in Cashu, we can even get to payment speeds almost as fast as data packages traveling through the network. The receiver Carol creates a public key and shares it with the sender Alice. Before sending money, the Alice prepares her ecash with one round-trip to the mint to lock it to the pub key of the receiver Carol. The sender Alice now has ecash that can only be spent by Carol (or by herself after a time lock). 

That way, we can send ecash from Alice to Carol over the network without the Carol having to make another round trip with the mint to know whether the ecash she received was already spent. If it is locked to her pub key (which she can immediately check) and if the mint has provided a valid signature (which she can check as well), the transaction is final.

 Ultrafast payments like these should take a couple of milliseconds if the network is fast enough.