The Karp--Lipton collapse

Let $R(x,y,z)$ be any polynomial-time predicate with polynomially bounded $z$, and form the NP language $E_R=\{⟨x,y,w⟩:\text{some extension }z⪰w\text{ satisfies }R(x,y,z)\}\mathrm{.}$ By the hypothesis, $E_R$ has polynomial-size decision circuits. Successively querying whether the current prefix has a satisfying extension, first with a $0$ appended and then with a $1$ appended, composes polynomially many copies of these decision circuits into a polynomial-size search circuit. On input $(x,y)$ for which some witness exists, it outputs an actual $z$ satisfying $R(x,y,z)$.

Let $L\in {\mathrm{\Pi }}_2^p$, so $x\in L⟺\mathrm{\forall }y\mathrm{\exists }zR(x,y,z)$. For each fixed input length, step 1.1 supplies a single polynomial-size search circuit $C$ which works for every pair $(x,y)$ of the relevant padded lengths. Therefore $x\in L⟺\mathrm{\exists }C\mathrm{\forall }yR(x,y,C(x,y))\mathrm{.}$ If $x\in L$, choose the search circuit from step 1.1; conversely, any circuit satisfying the right side supplies the required witness for every $y$. The predicate merely evaluates $C$ and checks $R$, so it is polynomial-time and the right side is a ${\mathrm{\Sigma }}_2^p$ formula.

Thus ${\mathrm{\Pi }}_2^p\subseteq {\mathrm{\Sigma }}_2^p$; the reverse containment after complementing gives equality of the second levels. Replacing each inner ${\mathrm{\Pi }}_2^p$ block of a higher alternating formula by its ${\mathrm{\Sigma }}_2^p$ form and merging adjacent existential and universal blocks inductively collapses every constant-alternation level to ${\mathrm{\Sigma }}_2^p$. Consequently $\mathrm{P}\mathrm{H}={\mathrm{\Sigma }}_2^p$.