Alessandro Chiesa [1] gave an accessible "State of the Snark" [2] presentation to the Silicon Valley Ethereum [3] meetup in October 2016 that is a good companion to this. Alessandro is working to implement SNARKs in such a way as to not require the trusted setup.

[1] https://people.eecs.berkeley.edu/~alexch/

[2] https://www.youtube.com/watch?v=g-HhDyXPWZc

[3] http://www.meetup.com/EthereumSiliconValley/

To follow up on the setup-less SNARKs, the idea is to create concretely efficient probabilistically checkable proofs. These are proof systems first pioneered ~25 years ago, and contain a lot of elegant mathematics but unfortunately to this day they are highly impractical. However, Ale has been doing a lot of work on bringing these systems into the realm of practical feasibility, and the next few years in this field will be super exciting.

The additional benefit of PCP-based SNARKs is that the cryptographic assumptions they rely on are pretty-well understood, and are almost standard today.

These are some pretty cool cryptographic primitives that form the basis for ZCash.

Thats a pretty long nutshell.