Explore a comprehensive lecture on the intersection of Trusted Execution Environments (TEEs) and blockchain technology. Delve into the fundamentals of TEEs, focusing on Intel's Software Guard Extensions (SGX) and their limitations. Examine partially decentralized systems, including Intel's Proof of Elapsed Time (PoET) and Proof of Useful Work. Investigate hierarchical attestation, performance considerations, and potential SGX compromises. Analyze the concept of payment channels, from unidirectional and bidirectional channels to the Lightning Network. Discover the innovative TEEChain system, its setup process, deposit mechanisms, and asynchronous blockchain access. Gain insights into crash fault resilience through persistent storage and chain replication. Conclude with a discussion on the global implications of blockchain technology for cross-border transactions.