This paper introduces MultiDimSPCI, a sequential conformal prediction (CP) method designed for multivariate time series forecasting. CP is a popular technique for uncertainty quantification in machine learning, known for its distribution-free and model-agnostic nature. MultiDimSPCI builds prediction regions in the form of ellipsoids, which are adaptively calibrated during the test phase to ensure valid coverage and small sizes. The method is theoretically grounded, providing finite-sample high-probability bounds on the conditional coverage gap without assuming exchangeability of the observations. Empirical results on simulated and real datasets demonstrate that MultiDimSPCI outperforms existing CP and non-CP methods in terms of both coverage and prediction region size, especially for high-dimensional time series. The paper also discusses the benefits of using local ellipsoids and compares MultiDimSPCI with copula-based CP methods, highlighting its advantages in handling non-exchangeable data and providing more precise uncertainty quantification.This paper introduces MultiDimSPCI, a sequential conformal prediction (CP) method designed for multivariate time series forecasting. CP is a popular technique for uncertainty quantification in machine learning, known for its distribution-free and model-agnostic nature. MultiDimSPCI builds prediction regions in the form of ellipsoids, which are adaptively calibrated during the test phase to ensure valid coverage and small sizes. The method is theoretically grounded, providing finite-sample high-probability bounds on the conditional coverage gap without assuming exchangeability of the observations. Empirical results on simulated and real datasets demonstrate that MultiDimSPCI outperforms existing CP and non-CP methods in terms of both coverage and prediction region size, especially for high-dimensional time series. The paper also discusses the benefits of using local ellipsoids and compares MultiDimSPCI with copula-based CP methods, highlighting its advantages in handling non-exchangeable data and providing more precise uncertainty quantification.