Biomarkers play a central role in medicine's gradual progress toward proactive, personalized precision diagnostics and interventions. However, finding biomarkers that provide very early indicators of a change in health status, for example, for multifactorial diseases, has been challenging. The discovery of such biomarkers stands to benefit significantly from advanced information processing and means to detect complex correlations, which quantum computing offers. In this perspective, quantum algorithms, particularly in machine learning, are mapped to key applications in biomarker discovery. The opportunities and challenges associated with the algorithms and applications are discussed. The analysis is structured according to different data types-multidimensional, time series, and erroneous data-and covers key data modalities in healthcare-electronic health records, omics, and medical images. An outlook is provided concerning open research challenges.