> ## Documentation Index
> Fetch the complete documentation index at: https://docs.qu.ai/llms.txt
> Use this file to discover all available pages before exploring further.

# Calculating Total Entropy

> How total entropy accumulates in Quai Network.

## From Single Blocks to Entire Chains

We've seen how PoEM measures the exact work in a [single block](/learn/advanced-introduction/poem/fork-choice/intrinsic-block-weight). But how do we compare entire blockchains? This is where **total entropy** comes in.

Think of it like this:

* **Single block**: "How hard was this math problem?"
* **Entire blockchain**: "How hard was this entire series of math problems?"

## Why Entropy Instead of "Work"?

**Entropy** is a physics concept that measures randomness or disorder. When miners find a block, they're essentially **reducing randomness** by finding a very specific, ordered hash.

**The Key Insight**: The amount of randomness removed (entropy reduced) directly corresponds to the energy spent creating that order.

**Real-world analogy**:

* Messy room (high entropy) → Clean room (low entropy)
* Energy required to organize = Entropy reduced
* More organized = More energy spent = Lower entropy

## Geometric vs Linear: Why It Matters

**Traditional PoW (Linear Addition):**

* Block 1: 16 difficulty points
* Block 2: 16 difficulty points
* Block 3: 16 difficulty points
* **Total**: 16 + 16 + 16 = 48 points

**PoEM (Geometric Multiplication):**

* Block 1: Removes 1/65536 of possible states
* Block 2: Removes 1/65536 of remaining states
* Block 3: Removes 1/65536 of remaining states
* **Total**: (1/65536) × (1/65536) × (1/65536) = Much more security

**Why Multiplication is Better:**

* **Exponential security**: Each block makes the chain exponentially harder to recreate
* **Real probability**: Reflects the actual odds of recreating the work
* **Better comparison**: More accurately measures which chain required more total energy

<Note>
  Think of it like compound interest: adding 10% three times gives you 130%, but compounding 10% three times gives you 133.1%. The difference becomes massive over many blocks.
</Note>

<Frame>
  <img src="https://mintcdn.com/dominantstrategies/X4sucm2SYhI37-ls/images/TotalEntropy.jpg?fit=max&auto=format&n=X4sucm2SYhI37-ls&q=85&s=9d3e23089ceb735b295e6a390e0ce7ed" width="1500" height="1055" data-path="images/TotalEntropy.jpg" />
</Frame>

## Step-by-Step Calculation

### Step 1: Calculate Single Block Entropy

For each block, we calculate how much randomness it removed:

**Simple Version**: More leading zeros = more randomness removed

**Precise Formula**:

```
Block Entropy = 1 / (2^leading_zeros)
```

**Example**:

* Block with 16 leading zeros: 1/65,536 states removed
* Block with 17 leading zeros: 1/131,072 states removed (twice as rare!)

### Step 2: Calculate Chain Total

For an entire blockchain, we multiply all the individual block entropies together:

**Formula**:

```
Total Chain Entropy = Block1 × Block2 × Block3 × ...
```

**Example 3-Block Chain**:

* Block 1: 1/65,536
* Block 2: 1/65,536
* Block 3: 1/131,072
* **Total**: (1/65,536) × (1/65,536) × (1/131,072) = 1 in 563 trillion

This means recreating this 3-block chain would require, on average, 563 trillion attempts!

## Practical Implementation: Using Logarithms

**The Problem**: Those numbers get huge fast! After just 10 blocks, we'd be dealing with numbers so large they're impossible to store efficiently.

**The Solution**: Instead of storing the actual multiplication results, we use logarithms to convert multiplication into addition.

**How It Works**:

* **Traditional storage**: 1/65,536 × 1/65,536 × 1/131,072 = 0.000000000000234
* **Logarithmic storage**: 16 + 16 + 17 = 49 bits of entropy

**Benefits**:

* **Manageable numbers**: Addition instead of astronomical multiplication
* **Exact precision**: No loss of accuracy in comparisons
* **Efficient storage**: Quai uses 64 bits to store total entropy per chain
* **Easy comparison**: Higher entropy number = more secure chain

**The Final Formula**:

```
Total Chain Entropy (in bits) = Block1_bits + Block2_bits + Block3_bits + ...
```

This mathematical trick allows Quai to precisely compare chains of any length while keeping the computation practical for real-world use.

<Note>
  **Key Takeaway**: PoEM measures the exact probability of recreating any blockchain, giving it perfect objectivity when choosing between competing chains. The chain that would be hardest to recreate always wins.
</Note>