Abstract: Vovk (2015) introduced cross-conformal prediction, a modification of split conformal designed to improve the width of prediction sets. The method, when trained with a miscoverage rate equal to $\alpha$ and $n \gg K$, ensures a marginal coverage of at least $1 - 2\alpha - 2(1-\alpha)(K-1)/(n+K)$, where $n$ is the number of observations and $K$ denotes the number of folds. A simple modification of the method achieves coverage of at least $1-2\alpha$. In this work, we propose new variants of both methods that yield smaller prediction sets without compromising the latter theoretical guarantee. The proposed methods are based on recent results deriving more statistically efficient combination of p-values that leverage exchangeability and randomization. Simulations confirm the theoretical findings and bring out some important tradeoffs.