Non-Interactive Verifiable Secret Sharing (NI-VSS) is a technique for distributing a secret among a group of individuals in a verifiable manner, such that shareholders can verify the validity of their received share and only a specific number of them can access the secret. VSS is a fundamental tool in cryptography and distributed computing. In this paper, we present an extremely efficient NI-VSS scheme using Zero-Knowledge (ZK) proofs on secret shared data. While prior VSS schemes have implicitly used ZK proofs on secret shared data, we specifically use their formal definition recently provided by Boneh et al. in CRYPTO 2019. The proposed NI-VSS scheme uses a quantum random oracle and a quantum computationally hiding commitment scheme in a black-box manner, which ensures its ease of use, especially in post-quantum threshold protocols. Implementation results further solidify its practicality and superiority over current constructions. With the new VSS scheme, for parameter sets (n, t) = (128, 63) and (2048, 1023), a dealer can share a secret in less than 0.02 and 2.0 seconds, respectively, and shareholders can verify their shares in less than 0.4 and 5.0 milliseconds. Compared to the well-established Pedersen VSS scheme, for the same parameter sets, at the cost of slightly higher communication, the new scheme is respectively 22.5× and 3.25× faster in the sharing phase, and notably needs 271× and 479× less time in the verification. Leveraging the new NI-VSS scheme, we revisit several classic and PQ-secure threshold protocols and improve their efficiency. Our revisions led to more efficient versions of both the Pedersen DKG protocol and the GJKR threshold signature scheme. We show similar efficiency enhancements and improved resilience to malicious parties in isogeny-based DKG and threshold signature schemes. We think, due to its remarkable efficiency and ease of use, the new NI-VSS scheme can be a valuable tool for a wide range of threshold protocols.