Sidechains have been widely used to improve the interoperability and scalability of blockchain systems. Despite several interesting sidechain constructions have been proposed in the literature, they suffer from the following downsides: (1) their designs do not easily support pluggable consensus mechanisms, and (2) their communication and storage costs for cross-chain operations are not yet optimized. In this work, we first propose Ge-Co, a generic sidechain construction to realize secure asset transfers between blockchains, supporting different consensus algorithms, such as Proof-of-Stake (PoS) and Proof-of-Work (PoW). Our design is built on top of the proposed voting committee selection approach and threshold signature schemes (TSS) and meanwhile, it achieves optimally succinct and constant proof size, only yielding lightweight communication and storage costs. Ge-Co works in the semi-adaptive corruption model. To provide stronger security, we further propose PoS-Co, a PoS-based sidechain construction in the fully-adaptive corruption model. PoS-Co is based on the proposed anonymous committee selection approach, and preserves optimally succinct proof. We also formally prove that Ge-Co can achieve the security properties of atomicity and timeliness. Finally, we develop a proof-of-concept (PoC) implementation for Ge-Co, and the results demonstrate that the design is efficient and practical.@en