The idea is to find an application that only has value if it is performed in a decentralized way. In my example the use of electricity to produce heat is a very good example. This has value in a decentralized setting, and is in fact very useful, but it is completely useless and worthless in a centralized setting. Why? Because one can transport the electricity for thousands of km but not the heat.
To consider human work. Typing a document can be done anywhere in the world and the digital document can be transmitted electronically, so this would not work. On the other hand, serving someone coffee only has value if it is done in the presence of the person served. By its very nature this kind of work has to be performed in a decentralized fashion, and could in principle be used to make proof of work useful.
1. (See the following quote, noting my red colorization.)
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2. Given that passports can be falsified, it would follow that any document representing something so trivial as “serving someone coffee” (ArcticMine) could be as well.