There exists an oracle A relative to which BQP is not contained in PH

You need to know: Notations for all $\forall$ and exists $\exists$ , computer program and number of operations it takes to run, quantum computer.

Background: Let $\Sigma=\{0,1\}$ , ${\Sigma}^n$ be the set of words $\sigma_1\dots\sigma_n$ with each $\sigma_i$ being $0$ or $1$ , and let $\Sigma^*=\bigcup\limits_{n=0}^\infty \Sigma^n$ . Let $|x|$ denote the length of any word $x\in \Sigma^*$ . Let $A:\Sigma^*\to \Sigma$ be an arbitrary function which we call oracle. Imagine a computer which can do everything a regular computer can do, but, in addition, can instantly compute $A(x)$ for any $x\in \Sigma^*$ . Let $P_k^A$ be the set of functions $F(x_1, \dots, x_k)$ in k variables $x_i\in \Sigma^*$ , for which there is a polynomial P and a program on such a computer which computes $F$ in at most $P\left(\sum\limits_{i=1}^k|x_i|\right)$ operations. Let $\text{PH}^A$ be the set of functions $f:\Sigma^*\to \Sigma$ for which there exists integer $k\geq 0$ , polynomial P, and function $F\in P_{2k+1}^A$ such that $f(x)=1$ if and only if $\forall y_1 \in \Sigma^{P(|x|)}\,\exists z_1 \in \Sigma^{P(|x|)} \dots \forall y_k \in \Sigma^{P(|x|)}\,\exists z_k \in \Sigma^{P(|x|)}$ such that $F(y_1,z_1,\dots,f_k,z_k,x)=1$ . Let $\text{BQP}^A$ be the set of all functions $g:\Sigma^*\to \Sigma$ for which there is a polynomial P and a program on a quantum computer (again with extra ability to instantly compute A) which computes $g(x)$ in at most $P(|x|)$ operations.

The Theorem: On 31st May 2018, Ran Raz and Avishay Tal submitted to the Electronic Colloquium on Computational Complexity a paper in which they proved the existence of oracle $A:\Sigma^*\to \Sigma$ such that $\text{BQP}^A \not\subseteq \text{PH}^A$ .

Short context: It is widely believed that quantum computers, when/if they will be build, will be able to efficiently solve a large class of problems than regular computers can do. More formally, it is believed that $\text{BQP} \not\subseteq \text{P}$ , and, moreover, $\text{BQP} \not\subseteq \text{PH}$ , where the definitions of P, PH, and BQP are as above but without oracle A. However, proving such separation results in well out of reach of the current techniques. Such problems often become easier with oracle, for example, it is not known whether $\text{P} \neq \text{PH}$ , but it is known that $\text{P}^A \neq \text{PH}^A$ for some oracle A. In 1993, Bernstein and Vazirani conjectured (at least verbally) the existence of oracle A such that $\text{BQP}^A \not\subseteq \text{PH}^A$ . The Theorem confirms this conjecture.