Bitcoin and computational complexity: a 'non collapsing' PH structure Bitcoin, as the most influential digital currency in the world today, is deeply rooted in the underlying assumptions of computational complexity theory due to its security and decentralized nature. Mapping the core mechanism of Bitcoin to the Polynomial Hierarchy (PH) in computer science can clearly reveal the subtleties of its design, especially in that each level maintains the characteristic of "difficulty", ensuring the "non collapse" of the entire system. PH Level: The Ladder of Computational Difficulties To understand the "non collapse" of Bitcoin, we first need to understand the PH hierarchy. It is a complex classification system for computational problems, built on the most basic classes of P (problems that can be solved deterministically in polynomial time, i.e. "easy" problems) and NP (problems whose solutions can be verified in polynomial time but are difficult to find). The PH level defines higher levels of complexity by alternately using "existential quantifiers" and "universal quantifiers". For example, NP problems can be simply understood as "Is there a solution; The more complex ∑ 2P problem may ask, "Is there an answer that provides a response to all possible rebuttals. In theory, this level can be infinitely extended, representing a continuous increase in computational difficulty. A core assumption of computational complexity theory is that there exists a strict containment relationship between these levels, namely P ⊊ NP ⊊ PH, which means that not all NP problems can be solved in P time, nor can all PH level problems be solved in NP time. If a high-level complexity class is equal to a low-level complexity class, we call it * * "collapse" * *. For example, if P=NP, then the entire PH hierarchy will collapse to the most basic P class, which means that all NP problems become "easy". Bitcoin's "three-layer" PH structure Bitcoin cleverly utilizes the difficulty of different problems in the PH hierarchy to construct a seemingly decentralized yet exceptionally robust system: Layer 1: Asymmetric elliptic curve encryption - In private key secure Bitcoin, users' funds are controlled by the private key. The generation of private keys is random, while deriving public keys and addresses from private keys is efficient (P problem). But deducing the private key from the public key or address is extremely difficult to compute. This is the fundamental security assumption of elliptic curve cryptography (ECC). The process of "inferring input from results" perfectly fits the characteristics of NP problems: given a private key, verifying its validity is easy; But there is currently no known efficient algorithm to find this private key. As long as the basic assumption of P!=NP holds, this security cornerstone of Bitcoin will never collapse to the P class, ensuring the privacy of user assets. Layer 2: Proof of Work (PoW) - The mining process of miners solving Nonce Bitcoin is the core of its consensus mechanism. Miners need to constantly try massive nonce values and perform hash operations to find a hash value that meets specific difficulty goals. This search process is computationally intensive, without any shortcuts, and can only be solved through large-scale trial and error. It is a typical NP problem: finding a suitable nonce is very difficult, but once found, any node can be verified in an instant (polynomial time). This "difficult to find and easy to verify" characteristic is the key to PoW mechanism preventing witch attacks and ensuring network fairness. Miners invest huge computing power to solve this problem, while other parts of the network can be verified with only a small amount of work, and this asymmetry constitutes the second layer of "non collapsing" barrier. Layer 3: Longest Chain Consensus - The Longest Chain Principle of Bitcoin is the cornerstone of consensus in distributed networks for network security and finality. At any given moment, there may be multiple valid blockchain branches in the network, but ultimately, all nodes will choose and extend the 'longest chain' that accumulates the most workload. Predicting which chain will become the longest chain in the future, or who will mine the next block, is almost impossible at the current point in time, involving complex factors such as randomness, computing power fluctuations, and network latency. This can be seen as a problem that is difficult to predict the future with a definite solution. However, verifying whether a given chain is currently the longest chain and checking whether the hash links of all blocks within it are valid is very easy and fast. This also presents the characteristics of NP problems: verifying the validity of a "solution" (specific chain) is simple, but "finding" or "predicting" the final "longest chain" is full of uncertainty. The emergence mechanism of distributed, multi verifier voting, as well as the convenience of verification and the difficulty of prediction, together constitute the third layer of the Bitcoin security model's "non collapsing" support. Conclusion: Robust PH structure Through the above analysis, we can see that the core design concept of Bitcoin, whether it is the security of private keys, the difficulty of proof of work, or the consensus mechanism of the longest chain, cleverly utilizes the characteristics of NP problems in computational complexity theory, namely "difficult to find and easy to verify". They all rely on "difficulties" that are currently not considered to belong to the P class, thus maintaining their inherent security and decentralization characteristics. Therefore, when it is said that 'Bitcoin's PH three-layer structure does not collapse', it means that its core security mechanisms rely on problems that are still computationally difficult. As long as the assumptions of complexity theories such as P!=NP hold true, these "difficulties" of Bitcoin cannot be easily cracked or reduced in dimensionality, thus ensuring the foundation for its long-term stable operation. This profound application of computational complexity theory is one of the key factors for Bitcoin to become such a robust and successful digital currency.