In this paper, we extend the techniques used in our previous work to show that there exists a probabilistic Turing machine running within time $O(n^k)$ for all $k\in\mathbb{N}_1$ accepting a language $L_d$ that is different from any language in $\mathcal{P}$, and then further to prove that $L_d\in\mathcal{BPP}$, thus separating the complexity class $\mathcal{BPP}$ from the class $\mathcal{P}$ (i.e., $\mathcal{P}\subsetneqq\mathcal{BPP}$). Since the complexity class $\mathcal{BQP}$ of {\em bounded error quantum polynomial-time} contains the complexity class $\mathcal{BPP}$ (i.e., $\mathcal{BPP}\subseteq\mathcal{BQP}$), we thus confirm the widespread-belief conjecture that quantum computers are {\em rigorously more powerful} than traditional computers (i.e., $\mathcal{P}\subsetneqq\mathcal{BQP}$). As an important consequence of the above results, we disprove the {\bf Extended Church-Turing Thesis}. Furthermore, we also show that (1): $\mathcal{P}\subsetneqq\mathcal{RP}$; (2): $\mathcal{P}\subsetneqq{\rm co-}\mathcal{RP}$; (3): $\mathcal{P}\subsetneqq\mathcal{ZPP}$. Previously, whether the above relations hold or not were long-standing open questions in complexity theory. Meanwhile, the result of $\mathcal{P}\subsetneqq\mathcal{BPP}$ shows that {\em randomness} plays an essential role in probabilistic algorithm design. In particular, we go further to show that (4): The number of random bits used by any probabilistic algorithm that accepts the language $L_d$ can not be reduced to $O(\log n)$; (5): There exists no efficient (complexity-theoretic) {\em pseudorandom generator} (PRG). $$ G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n; $$ (6): There exists no quick HSG $H:k(n)\rightarrow n$ such that $k(n)=O(\log n)$.

翻译：本文扩展了我们前期工作所使用的技术，首先证明存在一个对所有 $k\in\mathbb{N}_1$ 均在时间 $O(n^k)$ 内运行的概率图灵机，它接受一个与 $\mathcal{P}$ 中任何语言均不同的语言 $L_d$，进而证明 $L_d\in\mathcal{BPP}$，从而将复杂度类 $\mathcal{BPP}$ 与 $\mathcal{P}$ 分离（即 $\mathcal{P}\subsetneqq\mathcal{BPP}$）。由于有界误差量子多项式时间复杂度类 $\mathcal{BQP}$ 包含 $\mathcal{BPP}$（即 $\mathcal{BPP}\subseteq\mathcal{BQP}$），我们因此证实了量子计算机严格强于传统计算机这一广泛认可的猜想（即 $\mathcal{P}\subsetneqq\mathcal{BQP}$）。作为上述结果的重要推论，我们推翻了扩展丘奇-图灵论题。此外我们还证明：(1) $\mathcal{P}\subsetneqq\mathcal{RP}$；(2) $\mathcal{P}\subsetneqq{\rm co-}\mathcal{RP}$；(3) $\mathcal{P}\subsetneqq\mathcal{ZPP}$。这些关系是否成立此前曾是复杂度理论中长期悬而未决的问题。同时，$\mathcal{P}\subsetneqq\mathcal{BPP}$ 的结果表明随机性在概率算法设计中发挥着本质作用。进一步地，我们特别证明了：(4) 任何接受语言 $L_d$ 的概率算法所使用的随机比特数无法降低至 $O(\log n)$；(5) 不存在高效的（复杂度理论意义上的）伪随机发生器（PRG）：$$ G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n; $$ (6) 不存在快速的 HSG $H:k(n)\rightarrow n$ 使得 $k(n)=O(\log n)$。