This paper introduces two new conformal inference methods for improving the validity and quality of outputs from large language models (LLMs). The first method, conditional boosting, improves the scoring function used to filter claims by differentiating through the conditional conformal algorithm. The second method, level-adaptive conformal prediction, adjusts the claimed probability of correctness based on the characteristics of the prompt to retain more claims while ensuring validity. These methods address two key limitations of existing conformal inference approaches: the lack of conditional validity and the removal of too many valuable claims. The authors demonstrate the effectiveness of their methods on both synthetic and real-world datasets, showing improved claim retention and better calibration of the output probabilities. The results show that their methods provide more accurate and reliable guarantees for LLM outputs, particularly in scenarios where the model's performance may vary depending on the prompt's characteristics. The paper also provides theoretical guarantees for their methods, showing that they satisfy certain conditional coverage properties. The authors conclude that their methods enable practical applications of conformal prediction to LLMs, offering a way to ensure the validity of outputs while preserving their utility.This paper introduces two new conformal inference methods for improving the validity and quality of outputs from large language models (LLMs). The first method, conditional boosting, improves the scoring function used to filter claims by differentiating through the conditional conformal algorithm. The second method, level-adaptive conformal prediction, adjusts the claimed probability of correctness based on the characteristics of the prompt to retain more claims while ensuring validity. These methods address two key limitations of existing conformal inference approaches: the lack of conditional validity and the removal of too many valuable claims. The authors demonstrate the effectiveness of their methods on both synthetic and real-world datasets, showing improved claim retention and better calibration of the output probabilities. The results show that their methods provide more accurate and reliable guarantees for LLM outputs, particularly in scenarios where the model's performance may vary depending on the prompt's characteristics. The paper also provides theoretical guarantees for their methods, showing that they satisfy certain conditional coverage properties. The authors conclude that their methods enable practical applications of conformal prediction to LLMs, offering a way to ensure the validity of outputs while preserving their utility.