What are the consequences of $\mathsf{L}^2 \subseteq \mathsf{P}$?
The following is an obvious consequence:
$\mathsf{L}^{1+\epsilon} \subseteq \mathsf{P}$ would imply $\mathsf{L} \subsetneq \mathsf{P}$ and therefore $\mathsf{L} \neq \mathsf{P}$.

By the space hierarchy theorem, $\forall \epsilon &gt; 0: \mathsf{L} \subsetneq \mathsf{L}^{1+\epsilon}$ . 
If $\mathsf{L}^{1+\epsilon} \subseteq \mathsf{P}$ then $\mathsf{L} \subsetneq \mathsf{L}^{1+\epsilon} \subseteq \mathsf{P}$.