If $X\sim N(0,\Sigma)$ for some $d$-dimensional normal distribution, then $X = \Sigma^{1/2} Z$ where $Z\sim (0,I)$. How to compute the following quantity?
$$
\operatorname{var} (X^T X)
=
\operatorname{var}(Z^T \Sigma Z)
$$
Moreover, if we are in the growing dimension regime, under what condition posed on $\Sigma$ (such as restricted strong convexity) do we have as $d\to\infty$ the following?
$$
\operatorname{var}\left( X^\top X \right)
=
o(d^2)
$$


Update: do we have some sort of concentration inequality for $X^T X$?



Multivariate normal concentration
Let $ Y := Z^T \Sigma Z $. We have $ \operatorname{var}(Y) = \mathbb{E}(Y^2) - \mathbb{E}(Y)^2 $ and $ \mathbb{E}(Y) = \sum\_i \sigma\_{i, i} $. We thus need to compute $ \mathbb{E}(Y^2) $. For this, write
$$ \mathbb{E}(Y^2) = \mathbb{E}\left( \sum\_{i, j, k, \ell} Z\_i Z\_j Z\_k Z\_\ell \sigma\_{i, j} \sigma\_{k, \ell} \right) = \sum\_{i, j, k, \ell} \mathbb{E}\left(Z\_i Z\_j Z\_k Z\_\ell \right) \sigma\_{i, j} \sigma\_{k, \ell} $$
Due to $ \mathbb{E}\left(Z\_i \right) = 0 $, the only non zero contribution of this last expectation comes from the indices such that $ i = j \neq k = \ell $ (plus the permutations of this case) or $ i = j = k = \ell $. In the first case, we have $ \mathbb{E}\left(Z\_i^2 \right) = 1 $ and in the second case, $ \mathbb{E}\left(Z\_i^4 \right) = 3$. 


In the case $ \pi\_1 := (i = j \neq k = l ) $ the sum becomes $ (\sum\_{i \neq k} \sigma\_{i, i} \sigma\_{k, k} )^2 = (\sum\_i \sigma\_{i, i})^2 - \sum\_i \sigma\_{i, i}^2 $ and in the cases $ \pi\_2 := (i = k \neq j = l) $ and $ \pi\_3 := (i = l \neq j = k) $, it is equal to $ \sum\_{i \neq j} \sigma\_{i, j}^2 = \sum\_{i,j} \sigma\_{i, j}^2 - \sum\_i \sigma\_{i, i}^2 $. Summing, we get 
$$ \mathbb{E}(Y^2) = 2 \sum\_{i, j} \sigma\_{i, j}^2 + \left( \sum\_i \sigma\_{i, i} \right)^2 - 3 \sum\_i \sigma\_{i, i}^2 + 3 \sum\_i \sigma\_{i, i}^2 $$
hence, the variance is
$$ \operatorname{var}(Y) = 2 \sum\_{i, j} \sigma\_{i, j}^2 $$


Concerning your question on the regime $ d \to \infty $, you can for instance suppose that $ \max\_{i } \sigma\_{i, j}^2 = o(1) $ to get $\operatorname{var}(Y) = o(d) $ (with the trivial inequality $ \sum\_i \sigma\_{i, j}^2 \leq d \max\_i \sigma\_{i, j}^2 $). You can also put hypotheses directly on the quantity of interest, namely $ \sum\_{i, j} \sigma\_{i, j}^2 $.


