Earlier quoted context omitted.
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…
Is there a runnable, fully homomorphic encryption program that allows running turing complete languages?
IIRC it's a very fresh research. From what I heard It's definitely possible, but I don't think there are any working implementation of a turing complete machine on this.