Psoriasis: This is a common skin condition characterized by well-defined, red, scaly plaques. The description of quarter-sized plaques on the arms and legs is consistent with psoriasis, especially given its predilection for extensor surfaces.

Eczema (Atopic Dermatitis): Characterized by itchy, inflamed skin that can appear as plaques. While it more commonly affects flexural areas, it can also be found on extensor surfaces, especially in adults.
Seborrheic Dermatitis: A condition causing scaly patches, red skin, and stubborn dandruff due to an oily scalp. It can also affect oily areas of the body, such as the face, but less commonly the arms and legs.
Contact Dermatitis: An allergic or irritant reaction causing skin inflammation. This could result in plaques if the allergen or irritant is in contact with the skin on the arms and legs.

Cutaneous T-Cell Lymphoma (Mycosis Fungoides): A rare type of skin cancer that can mimic various dermatoses, including eczema or psoriasis. It's crucial to consider this diagnosis to avoid delayed treatment.
Sarcoidosis: A condition that can cause skin lesions, among other systemic symptoms. Skin plaques can be a manifestation, and missing this diagnosis could lead to untreated systemic disease.

Pityriasis Rubra Pilaris: A rare skin disorder characterized by salmon-colored scaling patches, which can sometimes appear as plaques.
Keratoderma: A group of rare skin conditions characterized by thickening of the skin on the palms and soles, but some forms can also affect other areas, leading to plaque-like lesions.
Lichen Planus: An inflammatory condition that affects mucous membranes inside the mouth and can also cause a skin rash. The rash is typically purplish, itchy, flat bumps, but it can occasionally present as plaques.