The Separation of $\mathcal{NP}$ and $\mathcal{PSPACE}$

There is a conjecture on $\mathcal{NP}\overset{?}{=}\mathcal{PSPACE}$ in computational complexity. It is a widespread belief that $\mathcal{NP}\neq \mathcal{PSPACE}$. In this paper, we show that $\mathcal{NP}\neq \mathcal{PSPACE}$ via the premise of $NTIME[S(n)]\subseteq DSPACE[S(n)]$, and then by diagonalization over all polynomial-time nondeterministic Turing machines via universal nondeterministic Turing machine $M_0$ running in $O(n^k)$ space for any $k\in \mathbb{N}_1$. Thus we obtain a language $L_d$ not accepted by any polynomial-time nondeterministic Turing machines, but accepted by $M_0$. We further prove that $L_d\in \mathcal{PSPACE}$ hence the result $\mathcal{NP}\neq \mathcal{PSPACE}$ follows.