Blockchains offer a useful abstraction: a trustworthy, decentralized log of totally ordered transactions. Traditional blockchains have problems with scalability and efficiency, preventing their use for many applications. These limitations arise from the requirement that all participants agree on the total ordering of transactions. To address this fundamental shortcoming, we introduce Charlotte, a system for maintaining decentralized, authenticated data structures, including transaction logs. Each data structurestructure -- indeed, each block -- specifies its own availability and integrity properties, allowing Charlotte applications to retain the full benefits of permissioned or permissionless blockchains. In Charlotte, a block can be atomically appended to multiple logs, allowing applications to be interoperable when they want to, without inefficiently forcing all applications to share one big log. We call this open graph of interconnected blocks a blockweb. We allow new kinds of blockweb applications that operate beyond traditional chains. We demonstrate the viability of Charlotte applications with proof-of-concept servers running interoperable blockchains. Using performance data from our prototype, we estimate that when compared with traditional blockchains, Charlotte offers multiple orders of magnitude improvement in speed and energy efficiency.