Actually apart from the concept of cryptocurrencies, the governments of different countries around the world are pretty excited about this technology.

In fact, many nations around the world have either started using blockchain or laid their plans for this technology in different areas of government administration or public affairs.

In my opinion, the events that would tell you the interest of the government in this technology would be

Europian union conducting the annual Blockathon an open event (competition) where engineers are invited to bring differently blockchain solutions that will help the government or businesses in different areas of administration.
Then the plan of Dubai (UAE) who are moving almost all the public service (taxation, wealth management, healthcare, business registration, etc.) on the blockchain by the end of 2020. So all of these services will be accessible within a few clicks via an online portal.

This article gives really interesting use cases of how different governments are using or planning to use blockchain technology in the future.

But let me tell you my favorite one, there is this small country in Europe (earlier it was the part of the soviet union) called Estonia. The country has a population of around 1.3 million people, and it is technologically the most advanced country I have ever seen.

What Dubi is planning to do they have done it way earlier. They have made 90% of the public service online using the blockchain technology. It including voting, health care & health records, taxation, vehicle purchase and registration, business registration and more.

They have been working on making the public service digital since 2008 even before blockchain was introduced, and unsurprisingly today they are the ones showing every other nation how you can use and adopt the technology.

For more cases: How countries around the globe are adopting blockchain for government administration

Watch the small documentary on Estonia by CNBC

https://youtu.be/NAxOUBMMQd4