What does $\Sigma^0_2$-hard and $\Pi^0_2$-hard for a TM's Acceptance Problem mean?
No, it's unrelated to NP-hardness.  $\Sigma_n^0$ and $\Pi_n^0$ are the levels of the arithmetical hierarchy.  $\Sigma_2^0$&nbsp;is the class of problems that can be decide by Turing machines that have an oracle for the halting problem and $\Pi_2^0$&nbsp;is the class of problems whose complement is in&nbsp;$\Sigma_2^0$.

There is the corresponding notion of the polynomial hierarchy, in which $\Sigma_1^\mathrm{P}$&nbsp;is NP and $\Pi_1^\mathrm{P}$&nbsp;is co-NP.