The complexity and black-box nature of LLMs makes it difficult to provide meaningful guarantees, which in turn complicates efforts to regulate and audit LLMs. In this talk, I will discuss how statistical guarantees on various properties of LLMs such as privacy (via differential privacy) and provenance (via watermarking or membership inference) provide powerful primitives for thinking about important regulatory issues such as copyright. At the same time, implementing and deploying these primitives can be challenging and I will discuss pitfalls and open problems in the interaction of various statistical guarantees and LLM deployment environments.