Which of the branches of accounting is primarily concerned with recording, classifying, allocating, and reporting current and prospective costs?

Cost Accounting

Explanation:

The branches of accounting that is primarily concerned with recording, classifying, allocating, and reporting current and prospective costs is cost accounting.

Cost Accounting is the collecting, recording, classifying, and appropriately allocating costs of an expenditure; and the categorization and reporting of all such costs either currently and prospectively.

Therefore a broader look at costing means that it entails

Collection of costs which are attributable to products or services

Classification of costs into material, labor or overheads

Recording the costs as they occur to avoid omission

Computation of total costs

Allocation of overhead costs to products and jobs

Reporting  the total costs for the period and by extension the standard costs for future periods