equivalent of a permissionless blockchain. Other works have considered anonymous credentials on blockchains[@cryptoeprint:2013:622; @coconut; @DBLP:conf/ccs/CamenischDD17; @quisquis; @indyAnonCreds], but requiring the issuance of new credentials and not reusing previously existing ones: verifying real-world identities and issuing their corresponding digital certificates is the most expensive part of any real-world deployment.

    -   The practical implementation and its perfomance evaluation \[sub:Performance-Evaluation\] for national identity cards and ePassports.

-   Circumventing the impossibility of full decentralization \[sub:Circumventing-the-Impossibility\] and the blockchain scalability trilemma.

-   A protocol for an incentive-compatible cryptocurrency \[alg:Incentive-Compatible-Protocol\]: previous blockchains mint cryptocurrencies tied to the process of reaching a consensus on the order of the transactions, but the game-theoretic properties of this mechanism is neither clear nor explicit.

-   A proof that mining the proposed cryptocurrency is a dominant strategy over other PoW/PoS blockchains and a Nash equilibrium over previous cryptocurrencies \[sub:Strictly-Dominant-Cryptocurrenci\], in addition to an Evolutionary Stable Strategy \[sub:Evolutionary-Stable-Strategies\].

-   The insight that the optimal efficiency of zk-PoI resides in that it’s implementing the social optimum, unlike PoW/PoS cryptocurrencies that have to pay the Price of (Crypto-)Anarchy \[sub:Price-of-Crypto-Anarchy\].

-   A proof that the circulation of the proposed zk-PoI cryptocurrency Pareto dominates other PoW/PoS cryptocurrencies \[sub:Pareto-Dominance-on\].

-   A proof that the proposed cryptocurrency could 