# Is $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?

Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?

• No, it’s not known. However, $\mathrm{BPP\subseteq MA\subseteq S_2P\subseteq ZPP^{NP}}$. Aug 8 '18 at 11:31
• Looks like an answer to me. Aug 10 '18 at 8:39

Heller gives a relativized world where $BPP = EXP^{NP}$ which sits far outside of $P^{NP}$. Showing $BPP \neq EXP^{NP}$ unconditionally would in itself be considered a major breakthrough in derandomization.
• What would $BPP\neq EXP^{NP}$ or $BPP\subseteq P^{NP}$ give in terms of lower bounds? Feb 23 '19 at 22:03