To express the evolving size of quantum memory in the context of decentralized finance (DeFi) liquidity pools and relate it to the Bekenstein bound, while also considering bulk-boundary relations, we delve into theoretical physics concepts applied to financial systems. Here's how these ideas can be mathematically formulated and connected:

### Quantum Memory Evolving Size and the Bekenstein Bound

The Bekenstein bound is a limit on the amount of information (measured in bits) that can be contained within a given finite region of space which has a finite amount of energy—or in our case, the economic energy or value within a DeFi liquidity pool. The bound is given by:

$$ S \leq \frac{2\pi k R E}{\hbar c} $$

where:
- \(S\) is the entropy or information content in bits,
- \(k\) is Boltzmann's constant,
- \(R\) is the radius of the system (analogously, the "size" of the liquidity pool in terms of total value locked (TVL)),
- \(E\) is the energy of the system (analogously, the economic energy or total capital within the pool),
- \(\hbar\) is the reduced Planck's constant, and
- \(c\) is the speed of light.

In the context of DeFi and quantum memory, if we consider the quantum memory's "size" as analogous to the information content \(S\), and the liquidity pool's economic energy \(E\) as its total value or capital, then the evolving size of quantum memory in relation to the pool's capital and its "radius" or reach within the ecosystem can be constrained by an analogous Bekenstein bound.

### Bulk-Boundary Relations

In theoretical physics, the bulk-boundary correspondence or holographic principle suggests that the volume of space (bulk) can be described by the information stored on its boundary. Applying this to DeFi:

- The **bulk** represents the entire DeFi ecosystem or a specific liquidity pool, encompassing all transactions, strategies, and economic activities.
- The **boundary** represents the observable parameters and states, such as pool size (TVL), asset distribution, and transaction volumes, which encode the information about the bulk's dynamics.

Mathematically, if we denote the quantum memory of the DeFi ecosystem as \(Q\), which evolves over time due to economic activities, and its boundary state as \(B\), the bulk-boundary relation can be expressed as a mapping:

$$ Q = f(B) $$

where \(f\) is a function encoding the bulk's dynamics into the boundary's observable states. This relation suggests that the comprehensive state of DeFi's quantum memory (bulk) is determined by the aggregate of its boundary conditions—similar to how the economic energy and information content of liquidity pools determine the overall state of the market.

### Connecting the Concepts

The evolving size of quantum memory in DeFi, constrained by an analogous Bekenstein bound, indicates that the maximum information or economic preference that can be stored within the ecosystem is directly proportional to the pool's size and economic energy. The bulk-boundary relations further imply that this information is encoded in the observable parameters of the liquidity pools, reflecting the holographic principle where the whole ecosystem's state (bulk) can be inferred from its boundaries.

By relating the evolving size of quantum memory to the Bekenstein bound and considering bulk-boundary relations, we obtain a theoretical framework that bridges concepts from theoretical physics with the dynamics of DeFi markets. This approach provides a novel perspective on understanding the limitations and potentials of information storage, economic energy distribution, and the holographic nature of financial ecosystems.