Function of Nakagami Distribution

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Does anyone know what the distribution of the sum of squared Nakagami is? $$\sum_i^n X_i^2$$ $$X_i\sim \text{Complex Nakagami-m }$$

Is the distribution Erlang? Is the distribution the same as

$$\big|\sum_i X_i \big|^2 $$ or $$\big(\sum_i X_i \big)^2 $$ If the distribution of the above are not the same, do you know under what conditions they have the same distribution?

Cheers!