Near-stateless, fully parallelizable validation of the Bitcoin blockchain with hints about which outputs remain unspent. All other inputs/outputs are efficiently crossed off inside a single hash aggregate that only reaches zero if validation was successful and the hints were correct.
15-minute talk summarizing the protocol
Validation is at the heart of Bitcoin. Any improvements in validation speed will have a direct impact on the scalability of the system, including everything that is built on top of it. For this reason improving validation performance may be one of the most important things we can do.