  Residual entropy, indicative of the degrees of freedom in a system at
absolute zero, is a cornerstone for understanding quantum and classical ground
states. Despite its critical role in elucidating low-temperature phenomena and
ground state degeneracy, accurately quantifying residual entropy remains a
formidable challenge due to significant computational hurdles. In this Letter,
we introduce the Temperature Incremental Monte Carlo (TIMC) method, our novel
solution crafted to surmount these challenges. The TIMC method incrementally
calculates the partition function ratio of neighboring temperatures within
Monte Carlo simulations, enabling precise entropy calculations and providing
insights into a spectrum of other temperature-dependent observables in a single
computational sweep of temperatures. We have rigorously applied TIMC to a
variety of complex systems, such as the frustrated antiferromagnetic Ising
model on both C60 and 2D triangular lattices, the Newman-Moore spin glass
model, and a 2D quantum transverse field Ising model. Notably, our method
surmounts the traditional obstacles encountered in partition function
measurements when mapping $d$-dimensional quantum models to $d+1$-dimensional
classical counterparts. The TIMC method's finesse in detailing entropy across
the entire temperature range enriches our comprehension of critical phenomena
in condensed matter physics. This includes insights into spin glasses, phases
exhibiting spontaneous symmetry breaking, topological states of matter and
fracton phases. Our approach not only advances the methodology for probing the
entropic landscape of such systems but also paves the way for exploring their
broader thermodynamic and quantum mechanical properties.
