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From the clue lengths and numbering we can deduce the grid to be as follows.

10d. Q₄ is 5 or 6 so Q₃₄ is 25 or 36. So 5a starts with a 5 or 6.

2d. Multiple of 25 or 36 with the multiple being 4, 5, 6 or 9. This yields

 
The only fit is with 150 so Q₁₀ is 6 and Q₃₄ is 25. Entering these in the grid gives up P₄ as 2. So we have P as ???2?????? and Q as **25*****6.

8a. Q₁₂ > P₁₂. So Q₁ is not 1.

3a. P₁₀ is 4, 5, 6 or 9. But from 8a it is not 4 and from the fact that P₄ is 2 it is not 5. So P₁₀ is 6 or 9. If it is 6 then Q₁₂ is 36 but Q₁₀ is 6 so P₁₀ is 9 and 3a is 81. Q₁ is 8 and Q₂ is 1. Using the 2 digit termini of square numbers tells us that P₉ is even but not 2 which already appears in P. Also Q₉ is odd and will be 3, 7 or 9.

8d. 6 x 81 = 486 so P₆ is 8 and P₈ is 6.

12a. P₂ or P₅ must be 5.

11d. 1 must be in P₁ to P₄ inclusive.

3d. This is a multiple of 12 so must be 84 and Q₈ is 7.

5a. Q₆ is 4.

7d. Starts with at least a 4 so 7a must be 90. Therefore Q₅ is 9, Q₉ is 3 and Q₇ is 0. Thus Q is 8125940736 which is 90144².

5a is 5940.

6d ends in 4 so P₅ is 4. Now P₉ is 0.

13a is ?60.

11d. P₂ must be 5.

4d is a multiple of 8 and is 104 so P₇ is 7.

12a is 35.

13a is 760.

11d is 30.

1d is 24 so P₃ is 3 and P₁ is 1.

P is thus 1532487609 which is 39147².

The remaining entries are

7d is 933, 9a is 3248, 1a is 213 and 6d is 914 which check out.

The final grid is as follows: