The blockchain exists to enable trustless transactions between peers. One of its applications is cryptocurrency. Another important one is smart contracts. In general, the idea is to eliminate the need for a trusted third party. Wherever lack of trust and the possibility of fraud is a problem, distributed ledgers can be of help.
One such area is supply chain management. In China, for example, counterfeit food products are a serious problem. A blockchain-based supply chain management system can involve meticulous time stamping and geolocating of products with cryptocurrency as the reward for all parties who take part in maintaining the integrity of the system. Using distributed ledger makes falsifying records later practically impossible.
The only way to implement electronic voting in a provably honest manner is to use the blockchain. With cryptography it is possible to even maintain privacy as a bonus while keeping the system provably fair.
In general, using the blockchain finding community consensus can be automated made verifiably correct. That may have uses in referenda.
But I believe it will take a long time before using the blockchain to radically alter the government is run. That's because vested interests will want to drag their heels to keep provably honest consensus mechanisms from being implemented as long as possible to keep the gravy train running.