Abstractly, the fast development of quantum computing is a threat to classical public-key cryptographic infrastructures. Asymmetric algorithms (which are widely used and include RSA and Elliptic Curve Cryptography (ECC)) are susceptible to the Shor algorithm, which can break them in a time-polynomial manner. This paper will give a detailed comparative study of Post-Quantum Cryptography (PQC) schemes founded on discrete algebraic structures such as lattice-based, code-based, hash-based, and multivariate polynomials based cryptosystems. These paradigms are systematically assessed according to computational complexity, key and ciphertext sizes, security reduction assumptions, and how they can be deployed on large scale networks. The NIST-standardized algorithms, which are the focus of our analysis, are ML-KEM (CRYSTALS-Kyber), ML-DSA (CRYSTALS Dilithium), FN-DSA (Falcon), and SLH-DSA (SPHINCS+). We also consider hybrid cryptography transitional measures to provide backward compatibility and security guarantee in the process of migrating the classical to quantum-resistant systems. As we found, the current state of lattice-based schemes provides the most desirable ratio of security to performance in large-scale applications, although, in the long run, code-based and multivariate schemes are necessary as their complements.