Quaternary sequences are the most important sequences in view of many practical applications. Using the reverse Gray mapping, we propose a new construction of elliptic curve quaternary sequences with strong pseudorandom properties. We apply a result of Marzouk and Winterhof on the relation between the pseudorandom measures for binary sequences and the measures for quaternary sequences to analyze the well-distribution measure and correlation measure of order, which were introduced by Mauduit and S′arkzy, for the proposed sequences. Quaternary sequences are the most important sequences in view many of practical applications. Using the reverse Gray mapping, we propose a new construction of elliptic curve quaternary sequences with strong pseudorandom properties. We apply a result of Marzouk and Winterhof on the relation between the pseudorandom measures for binary sequences and the measures for quaternary sequences to analyze the well-distribution measure and correlation measure of order, which were introduced by Mauduit and S’arkzy, for the proposed sequences.