Sparse Matrix Multiplication

You are given two matrices A and B represented in a sparse format. Each matrix is provided as a list of tuples, where each tuple is of the form (row_index, col_index, value) indicating the non-zero entries. In addition, you will be given the dimensions of each matrix:

• Matrix A has dimensions m x n
• Matrix B has dimensions n x p

Your task is to write a function that computes the product matrix C = A * B and returns it in the same sparse representation (i.e., a list of tuples containing only the non-zero entries).

Please ensure your function handles the following:

1. Validate that A and B have compatible dimensions for multiplication. If not, handle the error appropriately.
2. Consider the efficiency of your solution in terms of both time and space, keeping in mind that the inputs can be large but with mostly zero entries.
3. Manage edge cases such as one or both matrices having no non-zero entries.

Example Input:

Matrix A dimensions: 3 x 2
Matrix A entries: [(0, 1, 3), (2, 0, 4)]

Matrix B dimensions: 2 x 3
Matrix B entries: [(0, 2, 5), (1, 1, 6)]

Example Output:

Matrix C sparse representation: [ ... ]  
(Your function should compute the correct non-zero values for A * B in sparse form.)

Implement your solution in the programming language of your choice.