We study the output length of one-way state generators (OWSGs), their weaker variants, and EFIs. - Standard OWSGs. Recently, Cavalar et al. (arXiv:2312.08363) give OWSGs with $m$-qubit outputs for any $m=\omega(\log \lambda)$, where $\lambda$ is the security parameter, and conjecture that there do not exist OWSGs with $O(\log \log \lambda)$-qubit outputs. We prove their conjecture in a stronger manner by showing that there do not exist OWSGs with $O(\log \lambda)$-qubit outputs. This means that their construction is optimal in terms of output length. - Inverse-polynomial-advantage OWSGs. Let $\epsilon$-OWSGs be a parameterized variant of OWSGs where a quantum polynomial-time adversary's advantage is at most $\epsilon$. For any constant $c\in \mathbb{N}$, we construct $\lambda^{-c}$-OWSGs with $((c+1)\log \lambda+O(1))$-qubit outputs assuming the existence of OWFs. We show that this is almost tight by proving that there do not exist $\lambda^{-c}$-OWSGs with at most $(c\log \lambda-2)$-qubit outputs. - Constant-advantage OWSGs. For any constant $\epsilon>0$, we construct $\epsilon$-OWSGs with $O(\log \log \lambda)$-qubit outputs assuming the existence of subexponentially secure OWFs. We show that this is almost tight by proving that there do not exist $O(1)$-OWSGs with $((\log \log \lambda)/2+O(1))$-qubit outputs. - Weak OWSGs. We refer to $(1-1/\mathsf{poly}(\lambda))$-OWSGs as weak OWSGs. We construct weak OWSGs with $m$-qubit outputs for any $m=\omega(1)$ assuming the existence of exponentially secure OWFs with linear expansion. We show that this is tight by proving that there do not exist weak OWSGs with $O(1)$-qubit outputs. - EFIs. We show that there do not exist $O(\log \lambda)$-qubit EFIs. We show that this is tight by proving that there exist $\omega(\log \lambda)$-qubit EFIs assuming the existence of exponentially secure PRGs.

翻译：我们研究了单向状态生成器（OWSGs）、其弱化变体以及EFI的输出长度问题。 - 标准OWSGs。近期，Cavalar等人（arXiv:2312.08363）给出了对于任意$m=\omega(\log \lambda)$（其中$\lambda$为安全参数）可生成$m$量子比特输出的OWSGs，并猜想不存在输出为$O(\log \log \lambda)$量子比特的OWSGs。我们通过证明不存在输出为$O(\log \lambda)$量子比特的OWSGs，以更强的方式证实了该猜想。这意味着他们的构造在输出长度上是最优的。 - 逆多项式优势OWSGs。令$\epsilon$-OWSGs为OWSGs的参数化变体，其中量子多项式时间对手的优势至多为$\epsilon$。对于任意常数$c\in \mathbb{N}$，我们假设OWFs存在，构造了输出为$((c+1)\log \lambda+O(1))$量子比特的$\lambda^{-c}$-OWSGs。我们通过证明不存在输出至多为$(c\log \lambda-2)$量子比特的$\lambda^{-c}$-OWSGs，表明该结果几乎紧致。 - 常数优势OWSGs。对于任意常数$\epsilon>0$，我们假设次指数安全OWFs存在，构造了输出为$O(\log \log \lambda)$量子比特的$\epsilon$-OWSGs。我们通过证明不存在输出为$((\log \log \lambda)/2+O(1))$量子比特的$O(1)$-OWSGs，表明该结果几乎紧致。 - 弱OWSGs。我们将$(1-1/\mathsf{poly}(\lambda))$-OWSGs称为弱OWSGs。我们假设具有线性扩展的指数安全OWFs存在，构造了对于任意$m=\omega(1)$可输出$m$量子比特的弱OWSGs。我们通过证明不存在输出为$O(1)$量子比特的弱OWSGs，表明该结果是紧致的。 - EFI。我们证明不存在输出为$O(\log \lambda)$量子比特的EFI。我们通过证明在假设指数安全PRGs存在的情况下，存在输出为$\omega(\log \lambda)$量子比特的EFI，表明该结果是紧致的。