Current State Tree Structure:
Ethereum’s current Merkle Bridge Registry uses a hexary state tree structure. Nodes have 16 children, and each node is a hash of its child nodes. The state root is the root hash of the entire tree and is included in every Ethereum block. This structure allows for efficient verification of individual values within the state tree using Merkle branches.

Stateless Clients and Their Benefits:
Stateless clients aim to eliminate the need for nodes to store the entire state to verify Ethereum blocks. Stateless clients verify blocks by obtaining Merkle branches for the relevant state objects from other nodes. Stateless clients have several benefits, including reducing disk space requirements, easier node setup, and flexible validation options.

Challenges with Current State Tree Structure:
The sizes of Merkle branches in the current hexary state tree structure are too large for stateless clients. The length of a single branch is approximately 128 times the logarithm of the number of nodes in the tree. For a state tree with a size of 2^30 (approximately 1 billion nodes), the length of a branch is around 3.8 kilobytes. This size becomes a problem when dealing with blocks that contain a large number of transactions, leading to excessively large stateless blocks.

Potential Solutions:

1. Binary State Trees:
Transitioning from hexary to binary state trees can reduce branch sizes. With binary trees, only sibling nodes need to be included in a Merkle branch for verification, unlike in hexary trees, where all child nodes are required. This reduces the size of a single branch to approximately 64 times the logarithm of the number of nodes in the tree.

2. Vertical State Trees:
Vertical state trees are a more efficient variation of binary state trees. In vertical trees, nodes are arranged in a vertical fashion, with each node having two child nodes directly below it. This structure allows for more efficient Merkle branch generation, reducing the size of a single branch to approximately 32 times the logarithm of the number of nodes in the tree.

3. Cryptographic Accumulators:
Cryptographic accumulators, such as Merkle trees, can be used to efficiently prove the existence of an element in a set without revealing the entire set. They can be used to reduce the size of Merkle branches even further, making them more suitable for stateless clients.

4. SNARKs:
SNARKs (Succinct Non-Interactive Arguments of Knowledge) can be used to create zero-knowledge proofs of the validity of a state transition. This allows for the verification of state transitions without revealing the entire state, further reducing the size of Merkle branches.

Conclusion:
The transition to stateless clients in Ethereum requires a solution to the problem of large Merkle branch sizes. Various approaches, such as binary state trees, vertical state trees, cryptographic accumulators, and SNARKs, are being explored to reduce branch sizes and make stateless clients a reality. These solutions aim to improve the scalability, flexibility, and efficiency of Ethereum’s blockchain verification process.

Merkle Trees Efficiency:
Binary trees offer a significant improvement in efficiency over hex arrays in Merkle trees. By reducing the number of layers and sisters, binary trees result in proofs that are four times smaller and require less hashing.

Binary Tree Branch Length:
The branch length in a binary Merkle tree is reduced from 128 log n to 32 log n, resulting in a factor of four reduction in the cost of generating a branch.

Optimized Storage:
Storing data on disk in base 256 trees rather than binary trees allows for more efficient use of disk space. By storing lumps of 256 nodes in a single node, the database chunking size can be optimized, providing additional benefits.

Editing:
Editing in a binary Merkle tree involves hashing four times more nodes, but the hashes become eight times smaller compared to hex array Merkle trees. This trade-off results in overall efficiency gains.

Polynomial Commitments:
Polynomial commitments are an algebraic alternative to hash-based structures for committing to large amounts of data. Witnesses in polynomial commitments have a fixed size, regardless of the size of the polynomial being committed to. CATE commitments are a specific type of polynomial commitment scheme.

CATE Commitments for State Trees:
Using CATE commitments for state trees can reduce the proof size to just 48 bytes plus the data being proven. However, generating a proof for a CATE commitment requires polynomial division, which is an O(n) operation.

Pre-computing Witnesses:
Pre-computing all the witnesses in O(n log n) time can improve the proof generation time. However, this approach is not suitable for state trees because state trees are frequently read and written, requiring constant recomputation of witnesses.

Square Root of n Compromise:
To address the limitations of polynomial commitments for state trees, a square root of n compromise can be used. The state polynomial is split into an old state polynomial and a recent deltas polynomial. Witnesses for the old state polynomial are pre-generated and stored by clients. Witnesses for the recent deltas polynomial are generated in real time. When generating a proof, clients use a pre-generated proof against the old state and a proof against the recent deltas.

Benefits of the Square Root of n Compromise:
The cost of generating a branch is trivial for the old state and requires size square root of n arithmetic and elliptic curve work for the recent deltas. This approach allows for efficient proof generation while accommodating frequent reads and writes to the state tree.

Proof Generation Complexity:
Editing a branch in a binary tree requires square root of n log n cost, which is not ideal. Vertical trees have a tree of polynomial commitments, each with a size of 256 elements, resulting in a proof size of 12 log n bytes.

Proof Generation Process:
To generate proofs in a vertical tree, one needs to provide the intermediate polynomial commitment (48 bytes) and a proof (48 bytes) at each level. The process of generating a proof involves performing a size 256 linear combination, which can be replaced by editing values in place and recomputing commitments.

Editing Values:
Editing values in a vertical tree requires log n over 8 single elliptic curve multiplications, which is more complex than hashing but still efficient.

Further Improvement:
A clever protocol allows for providing a single proof that simultaneously proves multiple parent-child relationships, reducing the proof size further.

Proof Simplifications:
Replacing Kate commitments and BLS with bulletproof commitments and a regular 32-byte curve reduces proof size and mathematical complexity.

Benefits of Verkle Trees:
Reduced proof size by a factor of 32 compared to hex arrays. Simplified proof generation and verification. Lower computational costs for generating branches and editing proofs.

Alternative: Snarked Merkle Proofs:
Proof length consists of data plus a single SNARK. Editing cost is just hashing, while proof generation involves database writes and prover constraints. Requires proper verification, de-risking, and homework on arithmetically optimized hashes.

Drawbacks of Verkle Trees:
Not perfectly algebraic, lacking a clear relationship between nodes multiple levels down in the root. Requires zero-knowledge proofs over elliptic curve commitments and multiplications.

Conclusion:
Merkle trees are the current assumed path forward due to their simplicity and effectiveness. Alternatives like snarked Merkle proofs and starked Merkle proofs are viable options but require further research and development.

Virtual Trees:
Virtual trees are a data structure that combines properties of Merkle trees and polynomial commitments. They allow for efficient proofs of inclusion and exclusion, but may have some proof overhead. Virgal trees are currently the most promising design for this data structure, but there is still room for improvement.

Snarked Merkle Proofs:
Snarked Merkle proofs use a SNARK (succinct non-interactive argument of knowledge) to prove the validity of a Merkle proof. This eliminates the need to provide the Merkle branches, reducing the proof size. However, the SNARK itself is a proof that the witness exists, not the witness itself.

Proof Compression:
Proof compression techniques, such as multiproofs and efficient multiproofs, can be used to further reduce the size of Merkle proofs. Range proofs for Virgal trees are possible, but require a small amount of extra work compared to Merkle trees. Proofs of exclusion for Virgal trees are straightforward to construct.

Quantum Technology:
Quantum technology has the potential to break many cryptographic algorithms, including those used in blockchain technology. To prevent this, researchers are exploring post-quantum cryptography algorithms that are resistant to quantum attacks. Implementing these algorithms may require significant changes to existing blockchain protocols.

Quantum Computers and Cryptography:
Quantum computers can break certain types of cryptography, like RSA, class group based, and elliptic curve based, including pairings and simple elliptic curves. However, quantum computers do not break lattices and hashes.

Post-Quantum Merkle Trees:
Vertical trees are vulnerable to quantum computers, but snark-margle trees are not. A realistic post-quantum approach is to create a Merkle tree using an arithmetically friendly hash function and apply a Stark on top.

Resources for Further Reading:
Vitalik Buterin’s blog contains articles on Verkle trees and ZKSnarks. Dinkrad’s blog has a piece on Cat 8 commitments. ETH research post on the fancy scheme that doesn’t quite work (number 7520). Dinkrad’s piece on Cate commitments covers a lot of the math. Article discussing synarchs and polynomial commitments. A traditional Chinese article on the same topic. A picture explainer of how bulletproof commitments work, used in the vertical.