Recent decades have witnessed the prosperous development of the Internet of Things (IoT). To handle the bottlenecks of  the traditional cloud-based IoT framework, edge computing is widely introduced in IoT, to provide cloud services for terminal devices in a distributed and low-latency manner, forming a cloud-edge-terminal collaborative architecture. The heterogeneity, unreliability, and mobility of both IoT and edge devices raise great security and privacy concerns about cross-layer communications and collaborations, highlighting the necessity for a cross-layer authentication scheme. Existing authentication approaches suffer from heavy key management overheads and the need for a centralized authority. Besides, the current works mainly focus on terminal authentication while neglecting edge-terminal integrated authentication. To bridge these gaps, we propose a blockchain-based cross-layer IoT authentication scheme. Specifically, blockchain is utilized to construct trust among different entities and provide a platform where formatted transactions are broadcast and stored. Identity-based cryptography (IBC) is exploited for mutual authentication among edges and between edge and IoT devices. Experimental evaluation based on a prototype is conducted to show the effectiveness and efficiency of the proposed scheme.