Before revision:  Besides a direct k-pass implementation of classical spanner constructions ,\lfloor\rfloor the state of the art amounts to a O(\log k)-pass algorithm of Ahn, Guha and McGregor'12, and a 2-pass algorithm of Kapralov and Woodruff'14.
After revision:  Besides a direct k-pass implementation of classical spanner constructions ,\lfloor\rfloor+1-passes by Fernandez, Woodruff and Yasuda'20) the state of the art amounts to a O(\log k)-pass algorithm of Ahn, Guha and McGregor'12, and a 2-pass algorithm of Kapralov and Woodruff'14.