The Blockchain Trilemma refers to a theory that decentralized networks, such as blockchains, at any given time can only guarantee two of the following three benefits: decentralization, security, and scalability. Stated otherwise, any blockchain protocol must make design tradeoffs among the three potential benefits: decentralization vs. security vs. scalablity.
As of this writing there is no known blockchain protocol implementation that is decentralized, scalable, and secure.
Historical Note: The Blockchain Trilemma theory may be considered an evolution of the CAP Theorem in computer science, which posits that it is impossible for a distributed data network to guarantee more than two of the following benefits at any given time: consistency, availability, and partition tolerance.
The first of the following two videos introduces the concept of the blockchain trilemma. The second is an interview with Algorand founder Silvio Micali -- in particular, between 0:50 and 6:00, Micali describes the blockchain trilemma why he believes Algorand solves it:
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