[nosuchthingtm]

Consider a situation in which the US government realises that primecoin's output is crucial to further quantum computer research. They pump 100s of billions into mining primecoin. Now, 99.9999% of hashpower and full nodes belong to the US government - an entity whose only interest in primecoin is as a minor return on some work to reduce costs.

In this case, every assumption of trustlessness and censorship resistance has gone out the window.

A consensus algorithm using proof of useful work will become a market of miners for whom the mining rewards and thus the integrity of the network are merely a secondary concern.

[Retric]

Mining inherently means whoever is most efficient ends up with 51% of the market. If companies X spends less than anyone else per hash they will end up at 51% unless they intentionally hold back.

However, proof of useful work could be a public good. Though that makes coming up with an algorithm much harder.