In an idealized model, a longest-chain blockchain that weighs blocks by Gamma(S,V,W) is secure against private double-spending attacks if and only if Gamma is monotone and homogeneous of degree one in the timed resources V and W.
Proof of activity: Extending bitcoin's proof of work via proof of stake [extended abstract]y
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Nakamoto Consensus from Multiple Resources
In an idealized model, a longest-chain blockchain that weighs blocks by Gamma(S,V,W) is secure against private double-spending attacks if and only if Gamma is monotone and homogeneous of degree one in the timed resources V and W.