Threshold cryptography is essential for many blockchain protocols. For example, many protocols rely on threshold common coin to implement asynchronous consensus, leader elections, and provide support for randomized applications. Similarly, threshold decryption and threshold time-lock puzzles are often necessary for privacy. In this paper, we study the interplay between threshold cryptography and a class of blockchains that use Byzantine-fault tolerant (BFT) consensus protocols with a focus on latency. More specifically, we focus on blockchain-native threshold cryptosystem, where the blockchain validators seek to run a threshold cryptographic protocol once for every block with the block contents as an input to the threshold cryptographic protocol. All existing approaches for blockchain-native threshold cryptosystems introduce a latency overhead of at least one message delay for running the threshold cryptographic protocol. In this paper, we first propose a mechanism to eliminate this overhead for blockchain-native threshold cryptosystems with tight thresholds, i.e., in threshold cryptographic protocols where the secrecy and reconstruction thresholds are the same. However, many real-world proof-of-stake-based blockchain-native threshold cryptosystems rely on ramp thresholds, where reconstruction thresholds are strictly greater than secrecy thresholds. For these blockchains, we formally demonstrate that the additional delay is unavoidable. We then introduce a mechanism to minimize this delay in the optimistic case. We implement our optimistic protocol for the proof-of-stake distributed randomness scheme on the Aptos blockchain. Our measurements from the Aptos mainnet show that the optimistic approach reduces latency overhead by 71%.