Amorphous alloys have been severely restricted to widespread commercial application by their limited glass-forming ability (GFA). Since it is essentially a "freezing liquid" to some extent, studying the correlation of the atomic structures in solid with those in liquid during the rapid solidification of liquid alloys is expected to provide new insights for GFA. Therefore, the rapid solidification processes of liquid Ni₅₀Zr₅₀ alloy under six different cooling rates are simulated using molecular dynamics (MD) methods. The microstructure as well as its evolution of the rapidly solidified Ni₅₀Zr₅₀ alloys are characterized and analyzed with pair distribution function (PDF), cluster-type-index method (CTIM), and reverse atomic trajectory tracking method. The results show that the most numerous atomic configurations in Ni₅₀Zr₅₀ metallic glass is (11 2/1441 8/1551 1/1661), i.e., Z11 Kasper cluster, rather than icosahedra. The characteristic clusters in rapidly solidified Ni₅₀Zr₅₀ alloys tended to aggregate to form medium-range orders (MROs), and their numbers increase with the decrease of cooling rate. The configurational heredity of basic clusters emerges in the supercooled liquid region of T_m-T_g. Among the typical clusters, Z11 Kasper cluster possesses the highest fraction f of staged heredity in a wide temperature range above T_g. Raising cooling rate is beneficial for increasing f of basic clusters in the supercooled liquid region and increasing onset temperature of configuration heredity. The GFA of Ni₅₀Zr₅₀ alloy induced by raising cooling rate can be attributed to the enhanced hereditary ability of characteristic clusters such as Z11 Kasper cluster.