Koszul duality is an equivalence between rings and coalgebras. Topologically, a ring can be "delooped" through the bar construction into a coalgebra, simplifying its topological structure. Algebraic K-theory is a crucial invariant for rings that contains rich information but is exceptionally challenging to compute. Trace methods offer approximations of K-theory with more accessible ring invariants like Hochschild homology. In this talk, I show how these methods can be adapted to the coalgebraic context. Hess-Shipley introduced topological coHochschild homology as a similar invariant for coalgebras. I will present potential definitions for algebraic K-theory specific to coalgebras and provide corresponding trace methods. Additionally, I will show how these coalgebraic K-theories are compatible with the delooping of rings, rendering them valuable for computing the algebraic K-theory of rings.