Fully homomorphic encryption is a really big one. It allows to perform operations on an encrypted data - in theory you could run a computer program that would have it's memory contents encrypted at all times.

In case of cryptocurrencies it means that it's possible to create a cryptocurrency that has all the advantages of bitcoin, but also guarantees total anonymity - i.e. you're able to prove that you have the coins and not show where the coins come from. Mixers would not be necessary, and it would not be possible to track the coins.

The practical implementation of this is called DarkCoin. When we were doing Orisi we considered launching a sidechain to Bitcoin that would do similar stuff. But abandoned the project - privacy is important, but we all know who would be the first clients would be ;)

Also, it is be possible to create computer programs that run on your computer and have their own money, but that money is impossible to be stolen from them (the private key is never decoded and never reaches the computer memory).

Imagine a frustration of a computer hacker that gets a program which will pay him 1 BTC if, and only if he solves a specific equation. The program is open-source, and yet there provably is no way to hack into it.

Or a program that sends you money when it sees phrase "XXX" on Hacker News (you cannot cheat by providing altered HN website, because the program verifies HN SSL certificate on a homomorphically encrypted virtual machine).