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The Merkle Mountain Belt

Alfonso Cevallos, Robert Hambrock, Alistair Stewart

TL;DR

This work introduces Merkle Mountain Belt (MMB), a new family of Merkle-structure commitments designed for dynamic, append-only data in blockchains. By combining lazy merging, multi-layer bagging, and a belt layer, MMB achieves constant-size commitments with constant-time appends and polylogarithmic resync costs, while supporting incremental and asynchronous light-client interactions. The authors formalize increment proofs for append-only Merkle structures and show that MMB (and its variants U-MMB and F-MMB) deliver optimistic bounds on membership and increment proofs, especially for recency-biased queries. The framework offers practical benefits for light clients, cross-chain bridges, and stateless designs, reducing data-transfer and verification costs while maintaining strong security guarantees. Overall, MMB positions itself as the canonical Merkle structure for committed, growing data sets in blockchain applications, outperforming both Merkle chains and MMR in key dynamic-use cases.

Abstract

Merkle structures are widely used as commitment schemes: they allow a prover to publish a compact commitment to an ordered list $X$ of items, and then efficiently prove to a verifier that $x_i\in X$ is the $i$-th item in it. We compare different Merkle structures and their corresponding properties as commitment schemes in the context of blockchain applications. Our primary goal is to speed up light client protocols so that, e.g., a user can verify a transaction efficiently from their smartphone. For instance, the Merkle Mountain Range (MMR) yields a succinct scheme: a light client synchronizing for the first time can do so with a complexity sublinear in $|X|$. On the other hand, the Merkle chain, traditionally used to commit to block headers, is not succinct, but it is incremental - a light client resynchronizing frequently can do so with constant complexity - and optimally additive - the structure can be updated in constant time when a new item is appended to list $X$. We introduce new Merkle structures, most notably the Merkle Mountain Belt (MMB), the first to be simultaneously succinct, incremental and optimally additive. A variant called UMMB is also asynchronous: a light client may continue to interact with the network even when out of sync with the public commitment. Our Merkle structures are slightly unbalanced, so that items recently appended to $X$ receive shorter membership proofs than older items. This feature reduces a light client's expected costs, in applications where queries are biased towards recently generated data.

The Merkle Mountain Belt

TL;DR

This work introduces Merkle Mountain Belt (MMB), a new family of Merkle-structure commitments designed for dynamic, append-only data in blockchains. By combining lazy merging, multi-layer bagging, and a belt layer, MMB achieves constant-size commitments with constant-time appends and polylogarithmic resync costs, while supporting incremental and asynchronous light-client interactions. The authors formalize increment proofs for append-only Merkle structures and show that MMB (and its variants U-MMB and F-MMB) deliver optimistic bounds on membership and increment proofs, especially for recency-biased queries. The framework offers practical benefits for light clients, cross-chain bridges, and stateless designs, reducing data-transfer and verification costs while maintaining strong security guarantees. Overall, MMB positions itself as the canonical Merkle structure for committed, growing data sets in blockchain applications, outperforming both Merkle chains and MMR in key dynamic-use cases.

Abstract

Merkle structures are widely used as commitment schemes: they allow a prover to publish a compact commitment to an ordered list of items, and then efficiently prove to a verifier that is the -th item in it. We compare different Merkle structures and their corresponding properties as commitment schemes in the context of blockchain applications. Our primary goal is to speed up light client protocols so that, e.g., a user can verify a transaction efficiently from their smartphone. For instance, the Merkle Mountain Range (MMR) yields a succinct scheme: a light client synchronizing for the first time can do so with a complexity sublinear in . On the other hand, the Merkle chain, traditionally used to commit to block headers, is not succinct, but it is incremental - a light client resynchronizing frequently can do so with constant complexity - and optimally additive - the structure can be updated in constant time when a new item is appended to list . We introduce new Merkle structures, most notably the Merkle Mountain Belt (MMB), the first to be simultaneously succinct, incremental and optimally additive. A variant called UMMB is also asynchronous: a light client may continue to interact with the network even when out of sync with the public commitment. Our Merkle structures are slightly unbalanced, so that items recently appended to receive shorter membership proofs than older items. This feature reduces a light client's expected costs, in applications where queries are biased towards recently generated data.

Paper Structure

This paper contains 22 sections, 38 theorems, 52 equations, 15 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. U-MMB: an asynchronous scheme
  4. Append operation with lazy merges
  5. Resync operations and analysis
  6. Recent-proof compatibility
  7. F-MMB: a scheme with a constant-size commitment
  8. Bagging the peaks of a mountain range
  9. Append operation with forward bagging
  10. Resync operations and analysis
  11. The Merkle Mountain Belt
  12. Append operation with double bagging
  13. Resync operations and analysis
  14. The increment proof
  15. Conclusions
  16. ...and 7 more sections

Key Result

Theorem 1

MMB is an incremental, succinct and optimally additive commitment scheme. For $n=|X|$, it produces a constant-sized commitment, the manager needs $O(\log n)$ memory and $O(1)$ time per append, and a participant needs $O(\log n)$ memory and $O(\log k)$ CCC to perform any of the four basic resync oper

Figures (15)

  • Figure 1: A Merkle tree for list $X=(x_1, \cdots, x_8)$ and hash function $H(\cdot)$. For item $x_6$, its leaf $h_6$ is in green, and its membership proof is $\pi_{x_6 \in X}=((h_5, \text{left}), (h_{78}, \text{right}), (h_{14}, \text{left}))$, formed by the leaf's ancestors' siblings, in gray. From such an item and list, a verifier can reconstruct the actual ancestors, in brown, and ultimately the root, which should match public commitment $\langle X\rangle$.
  • Figure 2: In this Merkle chain, the root is in brown, and symbol $\bullet$ visually represents a non-existent child, and a default hash value for it. A new item $x_5$ can be appended to $X$ in constant time: add a new leaf $h_5=H(x_5)$, and create a new root $h'_{5}=H(h'_{4} || h_5)$ as the parent of $h'_{4}$ and $h_5$.
  • Figure 3: In $\mathop{\mathrm{\text{U-MMB}}}\nolimits$, membership proof $\pi_{x_i \in X_n}$ is valid against a commitment $\langle X_m \rangle$ for any value of $m$ in the shaded areas: the brown and green areas are guaranteed by old-commitment and recent-proof compatibilities, respectively.
  • Figure 4: $\mathop{\mathrm{\text{U-MMB}}}\nolimits$ structure for list $X=(x_1, \cdots, x_9)$, with height sequence $S_9=(2,2,0)$.
  • Figure 5: Evolution of the U-MMB structure from Figure \ref{['fig:s9']} through two item appends, where height sequence $S_9=(2,2,0)$ evolves to $S_{10}=(2,2,1)$ and to $S_{11}=(3,1,0)$. Peaks are represented with triangles. In each append, the new leaf is in green and the merge peak in gray.
  • ...and 10 more figures

Theorems & Definitions (67)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 57 more