We study the implications of the existence of weak Zero-Knowledge (ZK) protocols for worst-case hard languages. These are protocols that have completeness, soundness, and zero-knowledge errors (denoted $\epsilon_c$, $\epsilon_s$, and $\epsilon_z$, respectively) that might not be negligible. Under the assumption that there are worst-case hard languages in NP, we show the following:

1. If all languages in NP have NIZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+\epsilon_z < 1 $, then One-Way Functions (OWFs) exist.

This covers all possible non-trivial values for these error rates. It additionally implies that if all languages in NP have such NIZK proofs and $\epsilon_c$ is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition $ \epsilon_c+\sqrt{\epsilon_s}+\epsilon_z < 1 $ [Chakraborty et al., CRYPTO 2025].

2. If all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+(2k-1).\epsilon_z < 1 $, then OWFs exist.

3. If, for some constant $k$, all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ \epsilon_c+\epsilon_s+k.\epsilon_z < 1 $, then infinitely-often OWFs exist.