Sure, let's write a test case generator for the "Snake Escaping" problem. We'll need to generate random test cases based on the problem constraints:

1. An integer \( L \) (1 ≤ \( L \) ≤ 20).
2. An integer \( Q \) (1 ≤ \( Q \) ≤ 1,000,000).
3. A string \( S \) of length \( 2^L \) consisting of digits 0 through 9.
4. \( Q \) strings \( T_d \) of length \( L \) consisting of characters '0', '1', and '?'.

Here is the Python code for the test case generator:

```python
import random

def generate_test_case():
    # Generate L and Q
    L = random.randint(1, 20)
    Q = random.randint(1, 1000000)
    
    # Generate S of length 2^L with digits 0-9
    S_length = 2 ** L
    S = ''.join(random.choice('0123456789') for _ in range(S_length))
    
    # Generate Q strings T_d of length L with characters '0', '1', and '?'
    T_d_list = []
    for _ in range(Q):
        T_d = ''.join(random.choice('01?') for _ in range(L))
        T_d_list.append(T_d)
    
    return L, Q, S, T_d_list

# Function to print generated test cases in the required format
def print_test_case(L, Q, S, T_d_list):
    print(L, Q)
    print(S)
    for T_d in T_d_list:
        print(T_d)

# Example usage
if __name__ == "__main__":
    L, Q, S, T_d_list = generate_test_case()
    print_test_case(L, Q, S, T_d_list)
```

### Explanation:

1. **Generating \( L \) and \( Q \)**:
   - \( L \) is selected randomly between 1 and 20.
   - \( Q \) is selected randomly between 1 and 1,000,000.

2. **Generating \( S \)**:
   - The length of \( S \) is \( 2^L \).
   - Each character in \( S \) is randomly chosen from the digits '0' to '9'.

3. **Generating \( T_d \)**:
   - For each day \( d \), a string \( T_d \) of length \( L \) is generated.
   - Each character in \( T_d \) is randomly chosen from '0', '1', and '?'.

The `generate_test_case` function creates these values and the `print_test_case` function prints them in the format expected by the problem statement.

You can use this generator to create random test cases and validate your solution against different inputs.