Summary: For an arbitrary binary sequence of odd period \(p\) with optimal autocorrelation, quaternary sequences of period \(N=2np\) with three-level autocorrelation values are constructed, where \(n\) is an arbitrary positive integer and \(\gcd(2n, p)=1\). In particular, if \(n=1\), the quaternary sequence has optimal autocorrelation values. For a family of binary sequences of period \(p\) with optimal Welch bound, a family of quaternary sequences of period \(2p\) with almost optimal Welch bound is constructed.